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Secant Indices, Duality Defect, and Graeysnone S.r Joargleinzsoantions of the Segre Zeta Function

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COLLEGE OF ARTS AND SCIENCES

SECANT INDICES, DUALITY DEFECT, AND

GENERALIZATIONS OF THE SEGRE ZETA FUNCTION

By

GRAYSON S. JORGENSON

A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy

2020

Copyright c 2020 Grayson S. Jorgenson. All Rights Reserved. Grayson S. Jorgenson defended this dissertation on March 30, 2020. The members of the supervisory committee were:

Paolo Aluffi Professor Directing Dissertation

Svetlana Pevnitskaya University Representative

Ettore Aldrovandi Committee Member

Mark van Hoeij Committee Member

Kathleen Petersen Committee Member

The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements.

ii ACKNOWLEDGMENTS

I wish to thank Paolo Aluffi for all his guidance, encouragement, and support over the duration of my doctoral work. I always look forward to meeting with him and I have greatly enjoyed hearing his insights. His passion for math and for science and the arts as a whole has been an enormous inspiration to me, helping to drive my own work and curiosity. I am extremely grateful to have been able to work under his tutelage. Throughout my degree I felt fortunate to have been able to get to know many of the other great people in the FSU math department. In particular, I would like to thank the professors who taught me in a variety of courses in the earlier stages of my studies: Ettore Aldrovandi, Phil Bowers, Sergio Fenley, Wolfgang Heil, Alec Kercheval, Eric Klassen, Craig Nolder, and Mark van Hoeij. Thanks also to my fellow grad students, especially Arun, Ben, Braulio, Michael, Minfa, and Yi for many fun times and discussions. I would also like to express here my gratitude to Benjamin Hutz to whom I owe a large debt for playing a pivotal role in introducing me to the world of pure math during my undergraduate studies and for inviting me into several great opportunities over the years. Finally, I wish to thank my parents and sister for all of their love and support.

iii TABLE OF CONTENTS

List of Figures ...... vi Abstract ...... vii

1 Introduction 1 1.1 Secant indices ...... 1 1.2 Duality defect ...... 2 1.3 Relative Segre zeta functions ...... 5 1.4 Segre Zeta functions in the equivariant setting ...... 6

2 Secant Indices 9 2.1 Introduction ...... 9 2.2 General properties ...... 10 2.2.1 Strictness of growth ...... 12 2.3 Questions and a guiding principle ...... 16 2.4 Veronese varieties ...... 23 2.5 Segre varieties ...... 36 2.6 Lines on surfaces ...... 41

3 Linear Recurrence Sequences and the Duality Defect Conjecture 45 3.1 Introduction ...... 45 3.2 Preliminaries ...... 48 3.3 Dual varieties and duality defect ...... 49 3.4 A generalized algorithm and bounds for the degrees of counterexamples ...... 54 3.5 Duality defect in codimension 3 ...... 60

4 A Relative Segre Zeta Function 64 4.1 Introduction ...... 64 4.1.1 Motivation ...... 64 4.1.2 Statement of the results ...... 66 4.2 Proof of the main result ...... 70 4.3 Extension of bundles and sections ...... 75 4.4 The case of subbundles ...... 77 4.4.1 When F = E ⊕ L ...... 77 4.4.2 Reduction of the ambient space dimension ...... 79 4.5 Examples ...... 80 4.5.1 Subschemes of projective space ...... 80 4.5.2 Subschemes of products of projective spaces ...... 81

5 Equivariant Realization of the Segre Zeta Function 84 5.1 Introduction ...... 84 5.2 Segre classes ...... 86

iv 6 Conclusion 90

Bibliography ...... 92 Biographical Sketch ...... 96

v LIST OF FIGURES

(2) 2 2.1 Successive hyperplane sections of v3 (P )...... 25 2.2 Successive monomial additions to J for n = 2, d =4...... 26

vi ABSTRACT

This thesis consists of three independent projects in the field of algebraic geometry. The first of these is the focus of Chapter 2. There we define a family of integer sequences we n refer to as the family of secant indices of projective subvarieties. If X ⊆ P is a subvariety, its sequence of secant indices is denoted L(X) and records the maximal finite intersections of X with n linear subvarieties of P . We prove several properties about these sequences, develop a method to compute term-wise lower bounds and compute these lower bounds for Veronese and Segre varieties. We state several questions and conjectures regarding these sequences which to our knowledge are open. Chapter 3 concerns a project studying the so-called duality defect of projective subvarieties. n Given a projective subvariety X ⊆ P , one may consider its dual subvariety in dual projective space n ∨ (P ) . For low codimension smooth X, it would be a consequence of Hartshorne’s conjecture that the dual of X is always a hypersurface. This expectation is known as the duality defect conjecture, and in Chapter 3 we push a combinatorial approach of Holme and Oaland further to verify that the conjecture holds in the codimension 3 case for projective spaces of odd dimension, and also to derive a degree bound that any counterexamples would have to satisfy. Chapters 4 and 5 concern Segre classes of closed subschemes of projective bundles. By extending a given homogeneous ideal to progressively larger polynomial rings, one obtains an infinite sequence of closed subschemes of progressively higher dimensional projective spaces which are cones over the preceding subschemes. The Segre classes of these subschemes are related by the Segre zeta function defined by Aluffi. In Chapter 4, we generalize this Segre zeta function to subschemes of projective bundles over a smooth base variety. In Chapter 5 we explore a realization of the Segre zeta function in the context of equivariant algebraic geometry.

vii CHAPTER 1

INTRODUCTION

Here we summarize the content of the subsequent chapters while introducing some of the necessary prerequisite material.

First, some conventions we will fix throughout. We will take our base field to be k = C, though all the results of Chapter 4 hold when k is any algebraically closed field of characteristic zero. All schemes are assumed to be of finite type over k. A variety for us will always refer to an integral scheme, so in particular is assumed to be irreducible. All points are closed points.

1.1 Secant indices

n A fundamental observation in basic algebraic geometry is that if one has a subvariety X of P , n then for any hyperplane H ⊆ P , the dimension of X ∩ H is at least dim(X) − 1. Furthermore, it is always possible to find hyperplanes so that dim(X ∩ H) = dim(X) − 1 [32, Section I.7]. A n consequence of this is that if one has a linear subvariety L ⊆ P of dimension r, then dim(X ∩ L) ≥ dim(X) − n + r. Thus, if n − r ≤ dim(X), then the intersection X ∩ L must be nonempty. The case where dim(X) = n − r is special in that the smallest possible intersection X ∩ L will be a finite set of points. For “most” such L, X ∩ L will be finite. More precisely, the set of linear subvarieties of dimension r is parameterized by the Grassmannian G(r, n). When n − r = dim(X) there is a dense open subset of G(r, n) of linear subvarieties that all meet X in the same finite number of points. This maximal number of points that can appear in such an intersection is known as the degree deg(X) of X. By the same token, when dim(X) < n − r, then most L do not meet X at all; there is a dense open subset of G(r, n) of linear subvarieties with empty intersection with X. Despite this, nonempty intersections can still easily occur. A fundamental question is then to ask about the maximal possible number of points that can occur in such an intersection. Because most linear subvarieties of these smaller dimensions do not meet X, traditional intersection theory techniques

1 for addressing enumerative geometry problems cannot be used to produce exact answers. This question is an instance of a so-called quasi-enumerative geometry problem [20]. n In Chapter 2 we study this problem of special intersections. For a variety X ⊆ P and an integer 0 ≤ i ≤ codim(X) we define an integer

Li(X) := max{|X ∩ L|} where this maximum is taken over all dimension i linear subvarieties L with X ∩ L finite and reduced. We call the collection of these integers as a sequence

L(X) := (L0(X),..., Lcodim(X)(X)) the sequence of secant indices of X. We will derive several properties that these sequences satisfy. In particular, the sequence is always nondecreasing from left to right and the first and last numbers of the sequence are 1 and n deg(X), respectively. When X is not a minimal degree subvariety of P , that is, when deg(X) > codim(X) + 1, there must be at least one gap of size ≥ 2 in the sequence. Because of this, the sequences of secant indices for higher degree varieties seem to possess interesting combinatorial properties. We will investigate the cases where X is a Veronese or Segre variety in detail, obtaining term-wise lower bounds for L(X) that are conjecturally in fact equal to L(X). In the case of Veronese embeddings, the secant indices are related to the Cayley-Bacharach theorem and the open Eisenbud-Green-Harris conjecture.

1.2 Duality defect

N Given a variety X ⊆ P of dimension n, we say X is a scheme-theoretic complete intersection if the homogeneous ideal I(X) of X is generated by N − n homogeneous polynomials. A presently open conjecture was made over four decades ago by Hartshorne [31] that any smooth subvariety of N sufficiently small codimension in P should be a complete intersection. 2N More precisely, the conjecture states that a smooth X with n > 3 is a complete intersection. As of today, only partial progress toward this conjecture has been made. It is not even known whether all varieties of a fixed codimension ≥ 2 must be complete intersections when the dimension constraint is satisfied.

2 In Chapter 3 we study a related question concerning the duals of smooth subvarieties of pro- N jective spaces. Given a smooth n-dimensional subvariety X ⊆ P , one may define its dual in dual N ∨ N projective space (P ) . This dual projective space parameterizes the hyperplanes in P , and we ∨ N ∨ define the dual X ⊆ (P ) of X to be the collection of all “tangent hyperplanes” to X: those that contain at least one of the embedded tangent spaces of X. It is known that n ≤ dim(X∨) ≤ N − 1 always. The duality defect def(X) = N − 1 − dim(X∨) of X is designed to be a simple measure of how much X∨ fails to be a hypersurface. When X is a smooth complete intersection, it can be shown that X∨ is a hypersurface, and thus has duality 2N defect zero. Therefore, a reasonable conjecture to make is that when n > 3 , the duality defect of X is zero. A priori, this is a weaker version of Hartshorne’s conjecture. The conjecture has been attributed to Alan Landman and is referred to as the duality defect conjecture. This conjecture is notably weaker than Hartshorne’s conjecture in the sense that it has been fully proven in the codimension 2 case, see [15, Theorem 3.4] and [37, Corollary 6.4]. However, it still remains open in the higher codimension cases. One approach to proving the conjecture that we find particularly interesting is that of Holme [36] and his student Oaland [49].

To every scheme X one can associate the Chow group, A∗(X). This the quotient of the group L Z∗(X) = i≥0 Zi(X), where each Zi(X) is the free abelian group generated by the subvarieties of dimension i of X, by rational equivalence, see [23, Section 1.3]. Ai(X) denotes the corresponding i quotient of Zi(X) and we will often write A (X) := Adim(X)−i(X) when we wish to specify the groups by codimension. A proper map f : X → Y induces push-forward homomorphisms f∗ :

Ai(X) → Ai(Y ). When X is a smooth variety, there is an intersection product

Ai(X) × Aj(X) → Ai+j(X)

making A∗(X) into a ring sometimes denoted A(X) [23, Section 8.3]. For general schemes X, Y , flat morphisms f : X → Y induce pullback homomorphisms f ∗ : Ai(Y ) → Ai(X). Such pullback homomorphisms are defined also when f is only assumed to be a morphism between two smooth varieties X,Y [20, Theorem 1.23], and in this case they are also ring homomorphisms with respect to the intersection products defining A(X),A(Y ).

3 Given an algebraic vector bundle E on a scheme X, one may define the ith Chern class of E which is a homomorphism

ci(E): Ak(X) → Ak−i(X).

[23, Section 3.1]. Intuitively, once applied to the fundamental class of a variety, the Chern classes represent the rational equivalence classes of the loci where a set number of sections of E become linearly dependent, though to make this rigorous one must assume E has enough sections [20,

Section 5.3]. To denote ci(E) applied to an element α ∈ Ak(X) we write ci(E) ∩ α. The zeroth

Chern class is the identity endomorphism of A∗(X) and ci(E) is the zero map when i exceeds the rank of E. Thus one may formally invert the Chern polynomial

r ct(E) = 1 + c1(E)t + ... + crank(E)(E)t

dim(X) to obtain the Segre polynomial st(E) = 1 + s1(E)t + ... + sdim(X)(E)t of E. The map si(E): Ak(X) → Ak−i is called the ith Segre class of E. The combinatorial approach of Holme to address the duality defect conjecture is to consider ∨ the Segre classes of the bundle NX/PN (−1) , where NX/PN denotes the normal bundle of a smooth N N variety X in P . When applied to the fundamental class [X] and pushed forward to A(P ) via the N closed embedding i : X,→ P , the total Segre class s(E) ∩ [X] = (1 + s1(E) + ... + sn(E)) ∩ [X] becomes a class represented by an integer polynomial. The duality defect of X may be calculated from these integer coefficients and the coefficients in turn can be recursively expressed in terms of the Chern classes of NX/PN (−1). By capitalizing on numerical constraints that these Chern classes must satisfy, and the possibil- ities for homogeneous recurrence sequences with a prescribed set of zeros, in Chapter 3 we extend the work of Holme to prove that the duality defect conjecture is true for smooth codimension 3 sub- N varieties of P when N is odd. Along the way, using positivity properties of the Schur polynomials in these Chern classes, we prove also that if X is a smooth nonlinear codimension m subvariety of N 3N−2 P with positive duality defect r > 0 and N − m ≥ 4 , then

m j X N − m − r  deg(X) ≤ . 2 j=0

4 1.3 Relative Segre zeta functions

n For a closed subscheme i : Z,→ P , there is another type of Segre class, the Segre class of Z n n in P , s(Z, P ) ∈ A∗(Z). When Z is a smooth variety, this Segre class agrees with the one defined n n above for bundles in the sense that s(Z, P ) = s(NZ P ) ∩ [Z]. n n The Segre class s(Z, P ) contains information about the embedding of Z into P . While A∗(Z) is often a group without a simple presentation, by considering the pushforward

n n ∼ n+1 i∗s(Z, P ) ∈ A(P ) = Z[h]/(h ),

n one may still recover much information about s(Z, P ). Understanding the integer coefficients of n the lowest degree polynomial representing i∗s(Z, P ) is thus an important task. n A closed subscheme i : Z,→ P (x0 : ... : xn) corresponds to a homogeneous ideal I ⊆ k[x0, . . . , xn]. One can extend this ideal to a larger polynomial ring k[x0, . . . , xn] ⊆ k[x0, . . . , xN ], N N ≥ n. If we denote by IN the extended ideal, then we obtain a closed subscheme iN : ZN ,→ P N that acts as a cone over Z in P ; at least set-theoretically, Z is the image of ZN of the linear N n projection of P onto P away from the linear subvariety V (x0, . . . , xn).

In this way a given homogeneous ideal I defines an infinite sequence of cones ZN over Z.A reasonable question to ask then is whether the Segre classes of these subschemes are related. Recent work of Aluffi [5] answered this question by showing all these Segre classes could be obtained from a rational power series, known as the Segre zeta function ζI (t) of I. This function is defined to be the power series X j ζI (t) = ajt ∈ Z[[t]] j≥0 satisfying N ζI (HN ) = (iN )∗s(ZN , P )

N N for every N ≥ n, where HN ∈ A∗(P ) is the rational equivalence class of a hyperplane in P .

More specifically, if IN has homogeneous generators F1,...,Fr and if Xj denotes the closed subscheme cut out by Fj for each j, then we have the following fiber square:

N ZN P δ ∆ Qr Qr N j=1 Xj j=1 P

5 Qr N The normal bundle of the bottom regular embedding is a restriction of a bundle on j=1 P . N Denote by E the pullback of this bundle to P . Then tautologically we may always write

N N −1 N c(E) ∩ (iN )∗s(ZN , P ) (iN )∗s(ZN , P ) = c(E) ∩ c(E) ∩ (iN )∗s(ZN , P ) =: . c(E) ∩ [PN ]

The key point in showing that ζI (t) is a rational power series is showing that this “numerator”

N c(E) ∩ (iN )∗s(ZN , P ) has a fixed representation as an integer polynomial even as N increases. In Chapter 4 we define a relative version of this Segre zeta function that generalizes Aluffi’s Segre zeta function to describe the Segre classes of cones over closed subschemes of projective bundles with smooth varieties as bases. Let E,F be bundles over a smooth variety X, and suppose we have a surjection F → E of bundles. This surjection induces a rational map P (F ) 99K P (E). Given a closed subscheme i : Z,→ P (E) that is suitably defined, we can use this rational map to define the cone ˆi : Z,ˆ → P (F ) over Z in P (F ). The precise requirement is that Z is the zero scheme of a section s of a bundle G of rank < rank(E) on P (E), and there exists a sufficiently “compatible” bundle and section on P (F ). We define a formal power series with coefficients in A∗(X) denoted ζG,s(t) which is rational, equal to

P (t) , Q(t) for some polynomials P (t),Q(t) ∈ A∗(X)[t] depending only on the representations of the classes c(G) ∩ i∗s(Z,P (E)) and c(G) ∩ [P (E)] in A∗(P (E)). We show that this same rational expression describes the Segre class ˆi∗s(Z,Pˆ (F )) of Zˆ in P (F ) when such a cone Zˆ exists.

1.4 Segre Zeta functions in the equivariant setting

n Consider again the setup described above for a subscheme i : Z,→ P cut out by a homogeneous ideal I = (F1,...,Fr), where we have the fiber diagram:

n Z P δ ∆ Qr Qr n j=1 Xj j=1 P

6 As before, Xj denotes the zero scheme of Fj. One can define the scheme   n M n n+1 C := CZ P = Spec  I /I  n≥0

n n called the normal cone to Z in P [23, B.6]. Here I denotes the ideal sheaf of Z in P . By definition, n X j s(Z, P ) = p∗ c1(O(1)) ∩ [P (C)] j≥0 [23, Section 4.2], where   M n n+1 p : P (C) = Proj  I /I  → Z n≥0 is the projection map and O(1) denotes the hyperplane bundle on P (C). The normal cone embeds into the bundle N that is the pullback of the normal bundle of the bottom map in the fiber square to Z. If we denote by s : Z,→ N the zero section embedding, then ∗ there is a Gysin homomorphism s : A∗(N) → A∗(Z). Then by [23, Proposition 6.1 (a)], we see that ∗ n s [C] = {c(N) ∩ s(Z, P )}n−r, where [C] is the fundamental class of the normal cone C inside N, and the curly braces notation

{·}k specifies that we are taking the dimension k piece of the class within. n n If we push forward the class c(N) ∩ s(Z, P ) to A(P ) we will obtain the numerator of the n Segre zeta function for I. What this result is saying is that the term {c(N) ∩ s(Z, P )}n−r of the numerator corresponds directly to the fundamental class of the normal cone C in N. This term also n may be thought of as the intersection product X1 ·...·Xr of the divisors X1,...,Xr in P [5, Section 2]. In other words, the class s∗[C] only represents a single piece of the information of the numerator of ζI (t). In Chapter 5 we will examine the analogous situation in equivariant algebraic geometry ∗ ∗ where N is endowed with the fiberwise action of C and Z with the trivial C action. Then as it is a subcone of N, C is invariant under this action and therefore defines an equivariant fundamental class

∗ C [C]C∗ ∈ A∗ (E).

7 We will show there, in slightly greater generality, that this equivariant class of the normal cone captures the entirety of the information of the numerator of the Segre zeta function, rather than just that of one term.

8 CHAPTER 2

SECANT INDICES

2.1 Introduction

n Given a pure-dimensional closed subscheme X of P of codimension m, and a choice of integer n 0 ≤ j ≤ m, let Λj(X) ⊆ G(j, n) denote the subset of dimension j linear subvarieties of P with finite and reduced intersection with X. Then one may define

Lj(X) := max{|X ∩ L| | L ∈ Λj(X)}.

Here | · | is used to denote the set-theoretic count of the points of the scheme within. For our purposes, we refer to this integer Lj(X) as the jth secant index of X, and together these numbers form a sequence of length m + 1 starting at 1 and ending at deg(X), which we denote by L(X). Similar numbers have been studied independently, such as in [10] [48] [47] [43] [2]. It is known that the presence of a m-multisecant line, a line meeting X finitely in at least m points, implies that the Castelnuovo-Mumford regularity reg(X) of X is at least m [43, Proposition 1.1]. The cited references focus on classifying varieties with extremal secant subspaces with one of the goals being to provide examples of varieties with near maximal regularity. Such work provides evidence for the Eisenbud-Goto regularity conjecture [16] that reg(X) ≤ deg(X)−m+1 when X is a nondegenerate n codimension m subvariety of P . In this work, we are instead concerned with the properties that these indices satisfy collectively, as sequences. The sequences that occur in general seem to admit interesting combinatorial descrip- tions. For instance, given a smooth variety X, L(X) is always strictly increasing. Yet if X is not a n subvariety of minimal degree of P , then this sequence must contain gaps and these gaps need not occur only in one place in the sequence. One of the simplest examples of this is when X is the image 2 9 (2) 2 of the degree 3 Veronese embedding of P into P ; the sequence L(v3 (P )) is this case has 8 terms, (2) 2 but there are 9 numbers in the sequence 1, 2,..., 9. Here we have L(v3 (P )) = (1, 2, 3, 4, 5, 6, 7, 9) and the observation that the gap occurs between the final two terms of the sequence is tantamount to the classical Cayley-Bacharach theorem.

9 We attempt the development of a method to compute the secant indices for an arbitrary smooth variety that involves defining two accessory sequences to L(X), denoted by RLG(X) and RL(X), that are term-wise lower bounds for L(X). In the specific cases where X is a Veronese or Segre variety, we show that computing these accessory sequences is equivalent to solving two attractive combinatorial problems. These accessory sequences then produce conjectural values for L(X) for those varieties. In the case of Veronese varieties, we will show that the truth of the Eisenbud-Green- Harris conjecture [17] would imply the sequences RLG(X), RL(X) are indeed equal to L(X). The definitions of RLG(X) and RL(X) work for arbitrary smooth varieties, and the hope is that for every variety X there is a tractable combinatorial problem associated to computing these sequences, reflecting the nature of intersections of hyperplane sections of X which are as reducible as possible. This chapter is organized as follows. In Section 2.2 we describe properties that the sequence of secant indices satisfy in general, including the strictness of their growth in the case of a smooth n nondegenerate subvariety of P . We state a number of questions about the secant indices in Section 2.3 which to our knowledge are open, and define the integer sequences RLG(X), RL(X). In Section 2.4 we illustrate the accessory sequences and show they are equal when X is a Veronese variety. We prove a method that computes them and show that it would follow from the Eisenbud-Green-Harris conjecture that the sequences agree with L(X). In Section 2.5, we derive an algorithm to compute RLG(X) when X is a Segre variety and provide several computations. In Section 2.6 we discuss our original motivation of the problem of counting lines on surfaces, where our hope is that one can compute these maximal numbers by extrapolation from a related sequence of indices.

2.2 General properties

The most basic form of the main question we study in this chapter is as follows: what is the maximum number of points at which a linear subvariety of a given dimension can meet a pure- dimensional projective algebraic set X? This is only interesting when the linear subvariety has dimension small enough to meet X in finitely many points, thus we ask about the numbers

n max{deg(X ∩ L) | L ⊆ P linear, dim(L) = i, dim(X ∩ L) = 0}, for 0 ≤ i ≤ codim(X). The first of these, the maximum number of points that a single point can meet X, is clearly just 1.

10 However, for the other numbers there are a couple technicalities that require clarification. First, the exact method of counting the points in the intersections must be made precise. It would be nice to know that the last number in this sequence is always deg(X). If we count the points of a zero-dimensional scheme Y by letting deg(Y ) denote the scheme-theoretic degree rather than the set-theoretic count of the distinct closed points of the support of Y , this is not always the case.

4 Example 2.2.1. Consider the union X of two planes in P meeting at a single point. The degree of X is 2, but it is well-known that deg(X ∩ L) = 3 for any plane L meeting X at only its singular point.

To avoid more delicate issues like this, we opt to instead use the naive set-theoretic count, considering the numbers

n max{|X ∩ L| | L ⊆ P linear, dim(L) = i, dim(X ∩ L) = 0}, for 0 ≤ i ≤ codim(X). We still reserve deg(·) to denote scheme-theoretic degree, and instead denote by |Y | the cardinality of the set of closed points of a zero-dimensional scheme Y . The resulting sequence will always be nondecreasing.

Proposition 2.2.2. Suppose L is a linear subvariety of dimension < codim(X) with X ∩ L zero- dimensional. Then there exists a linear subvariety L0 of dimension dim(L) + 1 with X ∩ L0 zero- dimensional, and L ⊆ L0.

This version of the question fell out from our original motivation which is discussed in Section 2.6. One of the properties that seems reasonable to expect is that for nondegenerate X this sequence should in fact be strictly increasing. To prove such a result we introduce one last refinement: we require the intersections we are counting to be reduced, and we restrict our attention to smooth projective varieties. There is then no distinction between using the set-theoretic count or the scheme-theoretic degree. In this section and Sections 2.3, 2.4, and 2.5, the sequences sporting these additional properties are our objects of study.

Example 2.2.3. Consider the rational normal curve C that is the image of the Veronese map (1) 1 d vd : P ,→ P , a degree d smooth curve. The sequence L(C) has d terms, starting at 1 and ending at d. It is indeed the only strictly increasing sequence of that length connecting those two numbers,

11 L(C) = (1, 2, 3, . . . , d). Images of Veronese maps will be discussed further in Section 2.4. But the computation of the secant indices of rational normal curves can be dealt with much more easily: it d is well-known that any d + 1 points on C are linearly independent, that is, they span all of P . In particular, any r points on C are linearly independent, and so the most number of points a linear subvariety of dimension r can meet C is r +1. The intersection is reduced if it contains r +1 points.

0 2 2 Example 2.2.4. Suppose C is a smooth curve of degree d in P , and embed P as a plane in 3 3 P . Thus we obtain a degenerate degree d curve in P . It is clear that its secant indices are L(C) = (1, d, d). n More generally, if Y is a smooth subvariety of P , and Y is contained in a linear subvariety of dimension r, then the last n − r terms of L(Y ) are all deg(Y ). For another specific example, if Y is itself linear, then L(Y ) = (1, 1,..., 1).

2.2.1 Strictness of growth

n The sequence of secant indices of a smooth variety X in P is always nondecreasing. If X is nondegenerate, then the sequence is in fact strictly increasing. If X is degenerate, then L(X) has repeated terms as in Example 2.2.4, but is otherwise strictly increasing. We derive this property below using an elementary argument revolving around Bertini’s theorem. Note this argument is similar in essence to the elementary approach seen in [43] for deriving the upper bound on the secant indices, and indeed this upper bound is an immediate consequence of the strictness of growth we derive here. Many of the lemmas used below are well-known results, but for lack of appropriate references we give complete proofs for most of them here. The specific instance of Bertini’s theorem we will invoke throughout is the following.

Lemma 2.2.5. Suppose X is a smooth projective variety. Then a general element of a positive- dimensional linear system on X is smooth away from the base locus of the system.

As an immediate consequence, we have:

n Lemma 2.2.6. Let L be a linear subvariety and let X be a smooth subvariety of P . The set of all n n ∨ hyperplanes in P containing L forms a linear subvariety T of dual projective space (P ) . There is a nonempty open subset T of hyperplanes containing L and having smooth intersection with X outside of X ∩ L.

12 n For any closed subscheme Y of P (x0 : ... : xn) and a closed point p ∈ Y , we denote by Tp(Y ) the projective tangent space to Y at p. If Y = V (I) for a homogeneous ideal I with generators

F1,...,Fr ∈ k[x0, . . . , xn], then the tangent space is the linear subvariety cut out by the polynomials

∂Fi ∂Fi (p)x0 + ... + (p)xn, ∂x0 ∂xn for i = 1, . . . , r. The subscheme Y is smooth at p if and only if

dim(Tp(Y )) = dim(Y ) by the Jacobian criterion for singularities. Following immediately from this definition:

n Lemma 2.2.7. Let Y be a closed subscheme of P , and let L be a linear subvariety. Suppose p is a closed point of the scheme-theoretic intersection Y ∩ L. Then Tp(Y ∩ L) = Tp(Y ) ∩ L.

n Lemma 2.2.8. Let Y be a closed subscheme of P and let L be a linear subvariety of dimension < n − 1. Suppose L has reduced zero-dimensional intersection with Y , and the intersection consists n ∨ of the points p1, . . . , pr. Then if T is the linear subvariety of (P ) consisting of all hyperplanes containing L, there is an nonempty open subset of T of hyperplanes H with Y ∩ H smooth at the points p1, . . . , pr.

Proof. By hypothesis, dim(Tpi (Y )) = dim(Y ) for each i. Since Y ∩L is reduced, the points p1, . . . , pr are smooth points of Y ∩ L. By the Lemma 2.2.7 this means that for each i, the intersection

Tpi (Y ) ∩ L is a single point.

Fix an i. For a hyperplane H from T , the only way that Y ∩ H can be non-reduced at pi is n if H contains Tpi (Y ). The set of all hyperplanes in P that contain Tpi (Y ) is a linear subvariety n ∨ of (P ) , and so the set of all hyperplanes from T that contain Tpi (Y ) is a closed subset of T . It T must in fact be a proper closed subset, since L ∩ Tpi (Y ) is a single point, and H∈T H = L. Thus its complement in T is a nonempty open subset. The intersection of any finite collection of nonempty open subsets of a variety is nonempty and open, and so ends the proof.

Verifying that L(X) is strictly increasing is not difficult when X is a curve.

n Lemma 2.2.9. Let C be a nondegenerate, smooth, and irreducible curve in P . Then for any hyperplane H with C ∩ H reduced, the points of C ∩ H must span H.

13 Proof. Note we know that C ∩ H consists of exactly deg(C) points since C is nondegenerate and C ∩H is reduced. Suppose these points do not span H. Then their span is a proper linear subvariety n n L of H, so dim(L) ≤ n − 2. Then the set of all hyperplanes in P containing L covers all of P . Therefore, as C \ L is nonempty, we can choose a point p in C \ L, and there exists a hyperplane H0 containing L and also p. So C ∩ H0 consists of more than deg(C) points, which is impossible since C is nondegenerate.

n Lemma 2.2.10. Let C be a nondegenerate, smooth, and irreducible curve in P . Then L(X) is strictly increasing.

Proof. Let L be a linear subvariety with reduced intersection with C. If L is a hyperplane, there is nothing to show, so assume dim(L) < n − 1. n ∨ The set of all hyperplanes containing L is a linear subvariety of (P ) which induces a positive- dimensional linear system on C. By Bertini’s theorem and Lemma 2.2.8, there is a nonempty open subset of T consisting of hyperplanes with reduced intersection with C. We thus can pick such a hyperplane H. But by Lemma 2.2.9, the points of C ∩ L can only span at most L, which is properly contained in H. Thus C ∩ H must contain at least one additional point p.

If dim(L) = n − 2, then we are done; if L is taken to be linear subvariety realizing Ln−2(C), then we have shown Ln−2(C) < Ln−1(C). Otherwise, we can pick a linear subvariety L0 of dimension one greater than dim(L) and with 0 0 L ⊆ L ⊆ H, and p ∈ L . Thus Ldim(L)(C) < Ldim(L)+1(C).

The idea to get the general result is to reduce to the case of a curve when dealing with a higher dimensional variety. First, note that a reduced hyperplane section of a nondegenerate variety is nondegenerate, inside of the hyperplane, a sort of extension of Lemma 2.2.9.

n Lemma 2.2.11. Let X be a nondegenerate subvariety of P of dimension > 1 and suppose H is a ∼ n−1 hyperplane such that X ∩ H is reduced. Then X ∩ H is nondegenerate in H = P .

∼ n−1 Proof. Suppose to the contrary that there is a hyperplane L in H = P with X ∩ H ⊆ L. Since 0 n X is nondegenerate, there exists a point p ∈ X \ H. Thus we can find a hyperplane H of P containing both L and p. Further, H ∩ H0 = L.

14 The intersection X ∩ H0 is pure-dimensional of dimension dim(X) − 1, thus p is a point on an irreducible component Y of X ∩ H0 of that dimension. Then H ∩ Y must be a proper closed subset of Y , thus of smaller dimension, and therefore contained in one of the irreducible components of X ∩ H. 0 Let X1,...,Xr,Y,Z1,...,Zm be the irreducible components of X ∩ H , all considered with reduced scheme structures. Here X1,...,Xr are the irreducible components of X ∩ H. Since X ∩ H is reduced, we have

r X deg(X ∩ H) = deg(Xi), i=1 while

r m 0 X 0 0 X 0 deg(X ∩ H ) = mXi (X,H ) deg(Xi) + mY (X,H ) deg(Y ) + mZi (X,H ) deg(Zi). i=1 i=1

The notation mZ (A, B) stands for the intersection multiplicity of the intersection of two varieties A, B along an irreducible component Z of A ∩ B, as in [20]. This is a contradiction since the intersection multiplicities are positive and we must have deg(X∩ H) = deg(X ∩ H0), see [32, Theorem I.7.7].

n Theorem 2.2.12. Suppose X is a smooth nondegenerate subvariety of P . Then L(X) is strictly increasing.

Proof. By Lemma 2.2.10, we may assume dim(X) > 1. Suppose that L is a linear subvariety of dimension r < codim(X) so that X ∩ L is reduced and zero-dimensional, and |X ∩ L| = Lr(X). By Lemma 2.2.6 and Lemma 2.2.8, there is a hyperplane H containing L so that X ∩ H is smooth. The Fulton-Hansen connectedness theorem [24] implies any hyperplane section of X is connected, so the hypothesis that X ∩ H is smooth implies it is also irreducible. The previous lemma then shows X ∩ H is nondegenerate as a subvariety of H. Thus by induction, we may assume the existence of a linear subvariety T containing L of dimension n − dim(X) + 1 so that T ∩ X is a smooth, irreducible, nondegenerate curve in T . By Lemma 2.2.10, there exists a linear subvariety L ⊆ L0 ⊆ T of dimension r + 1 such that

|X ∩ T ∩ L0| > |X ∩ T ∩ L|.

15 Therefore

Lr+1(X) > Lr(X).

Note, as a minor consequence, this gives a slightly different way to think about the degree lower bound that all nondegenerate projective subvarieties satisfy, see for instance [19]. For smooth nondegenerate X, the fact that L(X) is strictly increasing forces deg(X) ≥ codim(X) + 1. That X is a variety is also essential. Secant indices of smooth, pure-dimensional, nondegenerate, and reduced subschemes do not necessarily form strictly increasing sequences.

3 Example 2.2.13. Consider the smooth curve C in P that is the union of three skew lines L1,L2,L3 all passing through another line L. Then L1(C) = 3, as L ∩ C consists of three distinct points, but any plane that contains L and meets C at points outside of L ∩ C must contain one of L1,L2,L3. So L(C) = (1, 3, 3) in this case.

One other immediate and basic consequence of the strictly increasing property is recovering the known upper bound for the cardinality of intersections with extremal secant spaces.

Proposition 2.2.14. Tautologically,

codim(X)−1 X Li(X) = deg(X) − (Lj+1(X) − Lj(X)), j=i for i = 0,..., codim(X) − 1. Thus in particular,

Li(X) ≤ deg(X) − codim(X) + i for each i.

See also Kwak [43]. When i = 1, this bound has been used as evidence for the Eisenbud-Goto regularity conjecture.

2.3 Questions and a guiding principle

n One of our main interests is a means of computing these numbers for a given subvariety X ⊆ P but this seems difficult for arbitrary subvarieties. It is clear that one method of obtaining a term- n wise lower bound for L(X) is to take a linear subvariety L ⊆ P of dimension codim(X) so that

16 X ∩ L is finite and reduced, and compute the sequence L(X ∩ L) where X ∩ L is considered as a ∼ codim(X) subscheme of L = P . r r r r First, for two integer sequences (aj)j=1, (bj)j=1 of the same length r, we write (aj)j=1  (bj)j=1 if aj ≤ bj for each j. This is a partial order on the set of all integer sequences of the same length. Additionally, we can define a total order on the set of integer sequences of the same length r as r r follows: (aj)j=1 ≤ (bj)j=1 if and only if either the sequences are equal, or there is a 1 ≤ k ≤ r such that ak < bk, and aj ≤ bj for every j > k.

n n Proposition 2.3.1. Let X ⊆ P be any smooth subvariety, and let H ⊂ P be any hyperplane not containing X so that X ∩ H is also smooth. Then

L(X ∩ H)  L(X)

∼ n−1 where X ∩ H is considered as a subvariety of H = P .

A natural question then is to ask when is this all that needs to be done.

n Question 2.3.2. For what subvarieties X of P is L(X) realized by L(X ∩L) for a linear subvariety L of dimension codim(X) such that X ∩ L is finite and reduced? Further, when is L(X) realized as a sequence of the form

(|X ∩ (H1 ∩ ... ∩ Hn)|,..., |X ∩ (H1 ∩ ... ∩ Hdim(X))|), for linearly independent hyperplanes H1,...,Hn?

Question 2.3.3. Consider Y := X ∩L for a linear subvariety L of dimension codim(X) with X ∩L reduced and finite. We can define two integer sequences:

1.

(max{|Y ∩ L0|},..., max{|Y ∩ Lcodim(X)|}),

2.

max{(|Y ∩ L0|,..., |Y ∩ Lcodim(X)|)}.

Here each Lj denotes a linear subvariety of dimension j contained inside L. The maximums in (1) are taken over all possible Lj, and the maximum in (2) is taken over all chains L0 ⊆ ... ⊆ Lcodim(X) using the total order ≤ defined above. Is it always the case that these two sequences are the same?

17 n Note that the answer to Question 2.3.3 is negative for arbitrary finite subsets Y of P .

Example 2.3.4. Let Y be a finite set consisting of 3 points p1, p2, p3 on a line T and 5 points 3 q1, . . . , q5 on a plane H in L = P . Suppose T is not contained in H, and p1, p2, p3 are not the point of intersection T ∩ H. Suppose also that the five points on H are arranged so that no three of them are collinear, and that no two of them lie on a line containing T ∩ H. Then the sequence from (1) of Question 2.3.3 is (1, 3, 5, 8), and the sequence from (2) is (1, 2, 5, 8).

At least for the case of Veronese varieties X considered in Section 2.4, the truth of the Eisenbud- Green-Harris conjecture would positively answer Question 2.3.2. In the same section we will also prove that the two sequences of Question 2.3.3 are equal, as a consequence of the Clements- Lindstr¨omtheorem. n For any finite set Y of points in P , computing L(Y ) is equivalent to considering the dimensions of the spans of all subsets of of Y , considered as subsets of points in the vector space kn+1. Another n question then becomes to ask for a subvariety X of P about what possible linear dependences between the deg(X) points of a reduced and finite intersection X ∩L where L is linear of dimension codim(X) occur as L is varied among all such linear subvarieties. One could phrase this in terms of matroids.

n Question 2.3.5. Let X ⊆ P be a variety. For each linear subvariety L of dimension codim(X) and X ∩ L finite and reduced, the subsets of linearly independent points of X ∩ L considered inside kn+1 form a matroid. What matroids can be realized in this way?

Finally, what sequences of integers can be realized as a sequence of secant indices?

n Question 2.3.6. If X is a smooth, nondegenerate subvariety of P , L(X) is a strictly increasing sequence of integers from 1 to deg(X) of length codim(X) + 1. What strictly increasing sequences of this length from 1 to deg(X) occur in this way?

Question 2.3.7. For X for which deg(X) exceeds codim(X) + 1, in what positions and in what sizes do the gaps in L(X) occur?

18 To our knowledge, the above questions have received little prior study, if any at all. For this last question, some of the related work on classifying varieties with extremal secant spaces provides a partial result about the existence of a gap between the last two terms of L(X). One such result is the following due to Kwak [43, Proposition 3.2].

n Proposition 2.3.8. Let X be a nondegenerate subvariety of P of dimension ≥ 1, and codimension ≥ 2. If X has an extremal curvilinear secant subspace of dimension 1,..., codim(X) − 1, then X is

5 1. a Veronese surface in P ,

4 2. a projected Veronese surface in P ,

3. a rational scroll.

As an immediate corollary of this result, we can classify all varieties that do not have a gap between the final two terms of L(·).

n Corollary 2.3.9. Let X be a nondegenerate smooth subvariety of P . Then unless X is a rational 5 4 scroll, the Veronese surface in P , or a projected Veronese surface in P ,

Lcodim(X)(X) − Lcodim(X)−1(X) ≥ 2.

Proof. The only work that needs to be done is reconcile our language with that used in the cited reference. A linear subvariety L is said to be a curvilinear secant subspace to X if X ∩ L if finite with each point of X ∩ L locally contained in a smooth curve on X. This last criterion is equivalent to specifying that dim(Tp(X) ∩ L) ≤ 1 for each point p ∈ X ∩ L. Such an L is called extremal if its intersection contains the maximal possible number of points, counted with appropriate multiplicity, length(X ∩ L) = deg(X) − codim(X) + dim(L). Our point of view in this chapter predominantly takes the more naive route of considering only reduced intersections; we only consider linear subvarieties L with X ∩ L reduced and finite. This ensures that dim(L ∩ Tp(X)) = 0 for each p ∈ X ∩ L and thus, in particular, such an L is a curvilinear secant subspace to X. Therefore, the result of Kwak implies that the bound seen in Proposition 2.2.14 is in fact strict for each i = 0,..., codim(X) − 1.

19 Kwak’s result along with similar work [48] on bounding the maximal possible lengths can be used in this way to treat the question of whether there is a gap between the final two terms of L(X) for all smooth nondegenerate varieties. But these results do not provide lower bounds for the terms of the sequence, and cannot be used to say more about the size of the penultimate gap, nor about the presence of other gaps in the sequence. Returning to the goal of computing L(X), a natural attempt to reduce the complexity of this computation is to compute the indices that arise when we only use a subset of the possible linear subvarieties. In particular, it seems reasonable to expect that the secant indices are the same if we were to only consider linear subvarieties cut out by hyperplanes that meet X in the most “reducible way” possible. This leads us to define two additional accessory sequences to L(X) which in some cases, such as those considered in Sections 2.4 and 2.5, become tractable to compute. To get a precise notion that generalizes beyond Veronese and Segre varieties we need to make several definitions.

n Definition 2.3.10. Let X be a smooth nondegenerate subvariety of P of dimension r.

• We say X is p-reducible if there exists a collection of hyperplanes H1,...,Hn so that their

common intersection is a single point, H1 ∩ ... ∩ Hr ∩ X is finite and reduced, each Hj ∩ X for j = 1, . . . , r is reduced and has exactly p distinct irreducible components, and finally

|H1 ∩ ... ∩ Hj ∩ X| > |H1 ∩ ... ∩ Hj+1 ∩ X| for each j = r, . . . , n − 1. Any such sequence of hyperplanes is said to satisfy the conditions of p-reducibility.

• We call the maximal p such that X is p-reducible the reducibility of X.

R • Suppose X has reducibility p. Denote by Λj (X) ⊆ Λj(X) the subset of linear subvarieties of dimension j cut out by hyperplanes H1,...,Hn−j such that the H1,...,Hn−j are the initial part of a sequence of hyperplanes satisfying the conditions of p-reducibility.

Note that by Bertini’s theorem, every smooth nondegenerate subvariety is at least 1-reducible.

n Proposition 2.3.11. Let X ⊆ P be a smooth nondegenerate subvariety. Then X is 1-reducible.

Proof. First, we can find hyperplanes H1,...,Hdim(X) so that each X ∩Hj is smooth and irreducible and so that H1 ∩ ... ∩ Hdim(X) ∩ X is finite and reduced by the classical Bertini theorem.

Let L = H1 ∩ ... ∩ Hdim(X) and note that the points of X ∩ L span L since X ∩ L is reduced so consists of deg(X) distinct points. By Lemma 2.2.8, if this were not the case, we would be able

20 to find a linear subvariety of dimension codim(X) meeting X at more than deg(X) points, despite the intersection being finite. Choose a subset of the points of X ∩L that span a linear subvariety L0 of dimension dim(L)−1.

Then by Lemmas 2.2.6, 2.2.8, we may find a hyperplane Hdim(X)+1 not containing L but containing 0 0 L so that X ∩ Hdim(X)+1 is smooth and irreducible. Since L has smaller dimension than L, we see that

|H1 ∩ ... ∩ Hdim(X) ∩ X| > |H1 ∩ ... ∩ Hdim(X)+1 ∩ X|.

Repeating this process completes the needed sequence of hyperplanes satisfying the conditions of 1-reducibility.

In general, it seems to be an independently interesting question about what the reducibility of a particular variety is. However, in the cases studied in Sections 4, 5, there is no mystery about n m the reducibility of the varieties in consideration. The reducibility of the Segre variety σ(P × P ) (n) n is 2 when n, m > 0, and that of the Veronese variety vd (P ) is d. At this point, one could define a new sequence which is a lower bound for L(X) at each term R by modifying the definition of Li(X) to only use linear subvarieties from Λ (X) instead of from Λ(X). This sequence seems interesting, but still appears challenging to compute. One difficulty in computing this new sequence is controlling how many irreducible components must be considered. To define the two accessory sequences to L(X) we will restrict the number of components.

n Definition 2.3.12. Let X ⊆ P be a smooth subvariety with reducibility p. Consider the set of all sequences of hyperplanes H1,...,Hn satisfying the conditions of p-reducibility. S • For each, consider the number of irreducible components in the union (Hj ∩ X) which have

nonempty intersection with H1 ∩ ... ∩ Hdim(X) ∩ X. Denote by µ(X) the minimal number of such irreducible components attained by the union of one of these sequences.

• Denote by H(X) the set of all sequences of hyperplanes (H1,...,Hn) satisfying the conditions S of p-reducibility and with the number of irreducible components of (Hj ∩ X), which each

have nonempty intersection with H1 ∩ ... ∩ Hdim(X) ∩ X, equal to µ(X).

Note that µ(X) ≥ p dim(X) always. If this were not the case, then this would contradict the p-reducibility properties of any sequence of hyperplanes H1,...,Hn realizing µ(X): either some of the hyperplane sections H1 ∩ X,...,Hdim(X) ∩ X would have to share an irreducible component

21 that meets the intersection H1 ∩ ... ∩ Hdim(X) ∩ X, which would violate the condition that H1 ∩

... ∩ Hdim(X) ∩ X is finite, or some of the Hj ∩ X would have to be nonreduced, for j ≤ dim(X). Once more, it seems to be an interesting question for arbitrary X what the value of µ(X) is. However, the for both the Veronese and Segre varieties we will consider, the minimal possible value p dim(X) is attained. We now are able to define the first of the accessory sequences to L(X).

Definition 2.3.13. Let X be as in the previous definition. Then if codim(X) = m, we define for each 0 ≤ j ≤ m an integer

n−j \ RLj(X) := max{|X ∩ L| | L = Hi, (H1,...,Hn) ∈ H(X)}, i=1 called the jth reducible secant index. As for the original secant indices, we denote these numbers collectively by

RL(X) := (RL0(X),..., RLm(X)).

n With this new terminology, for a given smooth subvariety X of P with reducibility p, we can now define the final sequence we are interested in.

Definition 2.3.14. We define the sequence of greedy reducible secant indices of X as

G RL (X) := max{(|X ∩ (H1 ∩ ... ∩ Hn)|,..., |X ∩ (H1 ∩ ... ∩ Hdim(X))|)}, where this maximum is taken over all (H1,...,Hn) ∈ H(X) and the sequences of integers are compared using the total order ≤ defined previously.

Altogether, we see that RLG(X)  RL(X)  L(X).

It seems conceivable that for many subvarieties X these three sequences are in fact equal. In Section 2.4 we will compute RLG(X), RL(X) when X is a Veronese variety, and will show they are equal. There we show also that the truth of the Eisenbud-Green-Harris conjecture would imply all three sequences are equal. With so little evidence however, we refer to the expectation in general that these sequences should all be equal as a “guiding principle” rather than a conjecture.

n Guiding principle. For any smooth nondegenerate subvariety X ⊆ P ,

RLG(X) = RL(X) = L(X).

22 2.4 Veronese varieties

In this section we consider the secant indices of the images of the Veronese embeddings. (n) (n) n Nd −1 n Throughout we will denote by vd : P ,→ P the degree d Veronese embedding of P into (n) (n) n+d (1) Nd −1
1 P , where Nd = n . In Example 2.2.3, we saw that L(vd (P )) = (1, 2, . . . , d) for every d > 0. (2) 2 5 Similarly, by Theorem 2.2.12, since the Veronese surface v2 (P ) has degree 4 in P , that is, it (2) 2 is a minimal degree subvariety, we see that L(v2 (P )) = (1, 2, 3, 4). The degree of the image of the (n) n Veronese embedding vd is d , and these sequences begin to become interesting when the degree exceeds the codimension of the image by more than one. (2) 2 The first case where this happens is for X = v3 (P ). This is a degree 9 subvariety of codi- 9 mension 7 in P . Thus L(X) has 8 terms, but there are 9 numbers in the sequence 1, 2, 3,..., 9. Therefore, the sequence of secant indices of X must contain exactly one gap, of size 2. The only remaining question is where in the sequence does this gap occur. In this case, the position of the gap is explained by the classical Cayley-Bacharach theorem [18].

2 Theorem 2.4.1. Let C1,C2 be two cubic curves in P not sharing any irreducible component and meeting in 9 distinct points. Suppose C3 is another cubic curve containing 8 of those 9 points.

Then C3 contains all 9 points.

Together with Theorem 2.2.12, this proves that L(X) = (1, 2, 3, 4, 5, 6, 7, 9). From this point of view, the content of the Cayley-Bacharach theorem is that the gap occurs between the final two terms of L(X). (n) n The correspondence in use here is that the hyperplane sections of vd (P ) are exactly the degree n d hypersurfaces in P . The maximum number of irreducible components such a hypersurface can (n) n have is d, and it is clear that the reducibility of vd (P ) is always d. G (n) n In what follows we give a method to compute RL (vd (P )) for all n, d > 1. We conjecture (n) n that this is in fact equal to L(vd (P )) in all cases. Some evidence in support of this is that the equality would follow from the truth of the currently open Eisenbud-Green-Harris conjecture, a conjecture which has received a large amount of attention in the last few decades admitting only partial progress [1] [12] [17] [28].

23 Our motivation in defining the sequences of greedy reducible secant indices becomes clear in G (n) n this context since the problem of computing RL (vd (P )) can be converted into a tractable combinatorial question. We illustrate the guiding heuristic to computing L(X) with the example (2) 2 X = v3 (P ).

9 Example 2.4.2. Consider two fully reducible hyperplane sections of X ⊆ P whose intersection 2 consists of exactly 9 distinct points. These hyperplane sections are curves in P (x : y : z), say

C1 = V ((x − z)(x − 2z)(x − 3z)) and C2 = V ((y − z)(y − 2z)(y − 3z)). We will compute RLG(X). The top term is 9, so then to find the next term we need to pick a 9 G hyperplane in P independent to the first two. The “greediness” of the sequence RL (X) is from the fact that we form it by finding hyperplane sections which remove the fewest points from the remaining finitely many points in the intersection at each step.

The number of irreducible components in the union C1 ∪ C2 is 6 = µ(X). To compute the next G term of RL (X) we must find an independent hyperplane section C3 which removes the fewest amount of points from C1 ∩ C2, but does not violate the µ(X) condition. That is, the way to interpret the defining constraint of the hyperplane sections used in the construction of RLG(X) is that they are those which do not introduce any additional irreducible components that meet the points we are working with; in other words, we must form C3 out of some choice of at most three distinct irreducible components from C1,C2.

These three components cannot all be from the same Cj, so it is clear that the best we can do is take, for example, C3 = V ((x − z)(x − 2z)(y − z)). Then C1 ∩ C2 ∩ C3 is reduced and consists of 7 points. Note also that so long as the number of points in the intersection decreases, the latest hyperplane section cannot be a linear combination of the previous. This process is continued to compute the remaining numbers. One possible continuation is illustrated in Figure 2.1. In Figure 2.1 the images are ordered left to right, top to bottom. The first image is of the 9 points in the intersection of C1 and C2. The curve C1 consists of the union of the three vertical lines, and the curve C2 of the three horizontal lines. These curves are all visualized by working 2 within the affine chart P \ V (z) and over the real numbers. The next image depicts C3 and the

7 points remaining after intersecting it with C1 ∩ C2. From there, the hyperplane sections chosen are V ((x − z)(x − 2z)z),V ((x − z)(y − z)(y − 2z)),V ((x − z)(y − z)z),V ((x − z)z2),V ((y − z)(y − 2z)z),V ((y − z)z2), in that order. Note the presence of the irreducible component V (z) in some of

24 (2) 2 Figure 2.1: Successive hyperplane sections of v3 (P ) these hyperplane sections, used to ensure the curve is of the correct degree. As V (z) does not meet

C1 ∩ C2, these additional hyperplane sections satisfy the conditions of d-reducibility.

This example was too simple to illustrate some questions that need to be resolved when com- puting RLG(·) in more complicated situations. One is, at a given step of the computation, if there are multiple possible new hyperplane sections to use, which each take away the same number of points, does it matter which one is used? Additionally, how can one determine whether the greedy sequence matches the sequence RL(·)? For Veronese varieties, answering these questions becomes easier if we first move the compu- tations to a simpler, more combinatorial setting. The heuristic guiding the transformation in this section is that the number of points we are getting after adding hyperplane sections is the same if we were to “collapse” the distinct irreducible components into one irreducible component, counted with a multiplicity. 3 So in the above example, for the purposes of the computation, we would collapse C1 into V (x ), 3 2 C2 into V (y ), and C3 into V (x y). The scheme-theoretic degree of their intersection is

3 2 2 deg(C1 ∩ C2 ∩ C3) = deg(V (x , y , x y)) = 7.

Therefore, a combinatorial problem we could consider is the following.

Combinatorial problem 2.4.3. Determine the maximal dimension of k[x0, . . . , xn−1]/I where I d d is the dehomogenization with respect to xn of a monomial ideal (x0, . . . , xn−1) + J in k[x0, . . . , xn] generated by r linearly independent degree d monomials, for each r ≥ n.

25 Example 2.4.4. We address this new problem for n = 2, d = 4. When r = 2, we obtain the 4 4 vector space k[x0, x1]/(x0, x1), of dimension 16. We see that the sequence of maximal dimensions is (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16), organized from r = 12 to r = 2. Figure 2.2 illustrates how this sequence can be obtained by sequentially adding monomials to the ideal J using a staircase diagram.

4 4 4 x1 x1 x1

3 3 3 x1 x1 x1

2 2 2 x1 x1 x1

x1 x1 x1

x0 2 3 4 x0 2 3 4 x0 2 3 4 x0 x0 x0 x0 x0 x0 x0 x0 x0

4 4 4 x1 x1 x1

3 3 3 x1 x1 x1

2 2 2 x1 x1 x1

x1 x1 x1

x0 2 3 4 x0 2 3 4 x0 2 3 4 x0 x0 x0 x0 x0 x0 x0 x0 x0

Figure 2.2: Successive monomial additions to J for n = 2, d = 4

This figure is read from left to right, top to bottom like Figure 2.1. The first image is that of the 4 4 16 monomials representing the basis elements of k[x0, x1]/(x0, x1). These monomials correspond to points in the diagram in accordance to their exponent vectors. Adding a monomial to J amounts to removing that monomial and all others it divides from the diagram; the number of monomials leftover is the dimension of the new polynomial ring quotient. We may only add monomials to J that are on or below the pictured diagonal line, as these are exactly those of degree ≤ 4. This setup leads to a “game” wherein one tries to remove as few monomials as possible from the diagram with each successive addition to J. The pictured sequence is part of the sequence of additions 3 3 2 2 2 2 3 2 3 2 2 x0x1, x0, x0x1, x0x1, x0, x0x1, x0x1, x0x1, x0, x1, x1, x1. The additions past x0 are not pictured as there are no more gaps larger than 1 that appear in the sequences of dimensions. Note that this sequence of monomials is part of the lexicographic sequence of degree 4 monomials in x0, x1, x2, but dehomogenized with respect to x2. A priori, this lexicographic sequence only gives a “greedy” sequence of removals, removing as few monomials from the diagram at each step as possible. This example is simple enough to check by brute force that this greedy sequence is in fact the term-wise

26 maximal sequence. That this is true for the more complicated examples when n or d is larger is more difficult.

G (n) n We will show that this combinatorial problem is equivalent to computing both RL (vd (P )), (n) n (2) 2 RL(vd (P )). In particular, in this example, we computed RL(v4 (P )). The two gaps observed in the sequence can be explained by a modern formulation of the Cayley-Bacharach theorem [17]:

r Theorem 2.4.5. Let Γ ⊆ P be a complete intersection of hypersurfaces X1,...,Xr of degrees 0 00 d1, . . . , dr, and let Γ , Γ ⊆ Γ be closed subschemes residual to one another. Set X m = −r − 1 + .

di Then for any ` ≥ 0, we have

0 r 0 r 1 r h (P , IΓ0 (`)) − h (P , IΓ(`)) = h (P , IΓ00 (m − `)).

As n and d grow, enumerating the possibilities for the dimensions of the involved cohomology groups becomes difficult, and so it does not seem feasible to use this result as a way of computing (n) n gaps in the sequences L(vd (P )). G (n) n (n) n We now show that the problem of computing RL (vd (P )), RL(vd (P )) is equivalent to computing the maximal dimensions of the polynomial ring quotients above. First we show that any dimension obtained there can be realized as the number of points left in the intersection of n hypersurfaces in P that are each the union of d hyperplanes. Qd Qd We know that the ideal I = ( i=1(x0 −ixn),..., i=1(xn−1 −ixn)) defines a complete intersec- n n tion in P . The support of this intersection is the collection of d distinct points {(a0 : ... : an−1 :

1) | ai ∈ {1, . . . , d}}, and so the complete intersection must be reduced. Adding any equation to n the set of generators of the ideal I gives an ideal which cuts out a reduced subscheme of P . Next, observe there is a divisibility-preserving bijection

R φn :Ωn → Ωn

R between Ωn and the set of monomials Ωn in k[x0, . . . , xn−1], defined

n−1 n−1 di Y di Y Y xi 7→ (xi − j), i=0 i=0 j=1 that is, for a, b ∈ Ωn, a divides b if and only if φn(a) divides φn(b). With this notation, we have the following key fact.

27 d d Lemma 2.4.6. Let I = (x0, . . . , xn−1, m1, . . . , mr) be a monomial ideal such that deg(mj) ≤ d for each j. Then

dimk k[x0, . . . , xn−1]/I

d d = dimk k[x0, . . . , xn−1]/(φn(x0), . . . , φn(xn−1), φn(m1), . . . , φn(mr)), as finite-dimensional vector spaces.

Proof. The dimension dimk k[x0, . . . , xn−1]/(φn(m1), . . . , φn(mr)) equals the degree of the reduced d d scheme V (φn(x0), . . . , φn(xn−1), φn(m1), . . . , φn(mr)). d d n We know that the support of the reduced scheme Y := V (φn(x0), . . . , φn(xn−1)) ⊆ A is the collection of dn distinct points

n {(a0, . . . , an−1) | ai ∈ {1, . . . , d}} ⊆ A

Qn−1 Qdi as discussed above. The effect of intersecting Y with a hypersurface of the form V ( i=0 j=1(xi − j)) is to remove all points (a0, . . . , an−1) of Y that satisfy ai > di for each i.

Qn−1 ai On the other hand, each point (a0, . . . , an−1) corresponds to a monomial i=0 xi . The quotient d d ring k[x0, . . . , xn−1]/(x0, . . . , xn−1) has as a basis the monomials corresponding to the points of Y .

Qn−1 Qdi The effect of intersecting Y with the hypersurface V ( i=0 j=1(xi − j)) is analogous to forming the quotient ring n−1 d d Y di k[x0, . . . , xn−1]/(x0, . . . , xn−1, xi ); i=0 the number of points left in Y is equal to the dimension of this new quotient ring. By repeating this process r times, we obtain the desired result.

Note the similarity between the patterns of Figures 1 and 2, accounting for the difference in dimension. Making rigorous this similarity is all that Lemma 2.4.6 is serving to do. The quotient (n) n ring problem leads to a clean proof that µ(vd (P )) = nd.

(n) n Lemma 2.4.7. µ(vd (P )) = nd.

(n) Proof. In the polynomial ring k[x0, . . . , xn−1] there are Nd monomials of degree ≤ d. d d n The quotient ring k[x0, . . . , xn−1]/(x0, . . . , xn−1) has dimension d as a k-vector space, and has d d as a basis the monomials from k[x0, . . . , xn−1] of degree ≤ d other than the pure powers x0, . . . , xn−1.

28 (n) It suffices to choose a sequence (aj)j of length Nd − n − 1 of these monomials so that aj does not divide ai for every i > j. This condition ensures that

d d dimk(k[x0, . . . , xn−1]/((x0, . . . , xn−1) + (a1, . . . , aj)))

d d > dimk(k[x0, . . . , xn−1]/((x0, . . . , xn−1) + (a1, . . . , aj+1))) for each j. N (n)−1 By Lemma 2.4.6, this sequence corresponds to a sequence of hyperplanes (H1,...,Hn) of P d so that d (n) n Y Hj ∩ vd (P ) = V ( (xi − ix0)) i=1

(n) n for each j = 1, . . . , n, and every other Hj ∩ vd (P ) for j > n is formed as the union of irreducible components of the preceding hyperplane sections, possibly along with the extraneous component (n) n V (xn) which does not meet H1 ∩ ... ∩ Hn ∩ vd (P ). Thus the union of these hyperplane sections has exactly nd irreducible components. Further, these hyperplanes all satisfy the requirements of d-reducibility.

G (n) n (n) n To compute RL (vd (P )), RL(vd (P )) we note a symmetry that occurs in these computa- tions.

(n) (n) n Nd −1 0 R Lemma 2.4.8. Let X = vd (P ) ⊆ P for n, d > 0. Then for any L, L ∈ Λcodim(X)(X), then for any sequence (H1,...,H (n) ) ∈ H(X) such that L = H1 ∩ ... ∩ Hn, there is a sequence Nd −1 0 0 0 0 0 (H1,...,H (n) ) ∈ H(X) with L = H1 ∩ ... ∩ Hn and Nd −1

0 0 |X ∩ (H1 ∩ ... ∩ Hj)| = |X ∩ (H1 ∩ ... ∩ Hj)| for each j ≥ n.

Proof. The key observation here is that since the reducibility of X is d and µ(X) = nd, each (j) (j) n hyperplane section Hj ∩ X for j = 1, . . . , n is a union of d distinct hyperplanes T1 ,...,Td in P .

The rest of the hyperplane sections Hj ∩ X are unions of the hyperplanes forming the preceding hyperplane sections. The intersection L ∩ X is finite and reduced, consisting of dn points. These points are exactly

Tn (jk) those of the form k=1 Tk where each 1 ≤ jk ≤ d.

29 0 n We may realize L ∩ X in the same way as an intersection of unions of hyperplanes in P . The formation of new hyperplane sections from unions of these constituent hyperplanes for the purpose of cutting away points from L ∩ X or L0 ∩ X can be done in the same way for both L and L0.

G (n) n (n) n Because of this symmetry, for the purposes of computing RL (vd (P )), RL(vd (P )) we may consider only sequences (H1,...,H (n) ) ∈ H(X) with Nd −1

d Y n Hj ∩ X = V ( (xj−1 − ixn)) ⊆ P , i=1 for j = 1, . . . , n. For any j ≥ n, we have that

|X ∩ (H1 ∩ ... ∩ Hj)| = deg(X ∩ (H1 ∩ ... ∩ Hj)) = dimk(k[x0, . . . , xn−1]/I) where I is the homogeneous ideal of X ∩ (H1 ∩ ... ∩ Hj) dehomogenized with respect to xn. By Lemma 2.4.6, we achieve our objective:

Lemma 2.4.9. We have

1.

G (n) n RL (vd (P )) d d = max{(dimk(k[x0, . . . , xn−1]/(x0, . . . , xn−1, a1, . . . , a (n) )), Nd −n−1 d d . . . , k[x0, . . . , xn−1]/(x0, . . . , xn−1))},

2.

(n) n RL (n) (vd (P )) Nd −n−1−j d d = max{dimk(k[x0, . . . , xn−1]/(x0, . . . , xn−1, a1, . . . , aj))},

d d where both maximums are taken over all monomials ak of degree ≤ d and so that x0, . . . , xn−1, a1, . . . , ak are all linearly independent, and the first maximum is done using the total order ≤ on integer sequences.

The question of whether these two sequences of secant indices are equal is then equivalent to asking whether for the integer sequences over which the maximum is taken in the expression for

30 G (n) n RL (vd (P )) above, is a maximal element with respect to the total order ≤ also a maximal element with respect to the partial order ? Our reduction of the problem to the quotient ring setup helps with the visualization of the combinatorics behind this question. In the n = 2 case, as in Example 2.4.4, the diagram is simple enough that the expectation that these two sequences are the same seems reasonable. But this is a fact which requires a careful proof. We can think of the combinatorial problem of computing the maximal dimensions of these quotient rings as a specific instance of a more general family of problems. Consider a poset which is decomposed into the disjoint union of two finite sets C = A ∪ B. Denote by ≤ the partial order on C and let m = |A|. The problem is to consider all sequences of length m−1 obtained by picking an element of A, and then removing it and all elements greater than it according to ≤ from C, subject to the condition that the chosen element does not remove any further elements from A.

That is, one considers all sequences (a1, . . . , am−1) of distinct elements of A such that aj 6≤ ai for all i > j, and then considers the sequence of cardinalities obtained by removing a1, . . . , am−1 one at a time, in that order, along with all elements larger than each. In our computation of the secant indices, C is the set of monomials that represent the generators d d of the quotient k[x0, . . . , xn−1]/(x0, . . . , xn−1), and the set A is that consisting of all monomials of d d degree ≤ d aside from the pure powers x0, . . . , xn−1 in k[x1, . . . , xd]. It is simple to construct examples of such posets in general where the greedy sequence is not maximal with respect to .

Example 2.4.10. Consider the poset {1, a, b, b2, c, a2, a2, a3, a4, a5}, where α ≤ β if and only if α divides β. Let A = {1, a, b, c, a2, a3}, and let B = {b2, a4, a5}. Choosing the elements of A in the order c, b, a3, a2, a yields the “greedy” sequence (9, 8, 6, 3, 2, 1). Whereas removing the elements in the order a3, a2, a, c, b yields the sequence (9, 6, 5, 4, 3, 1), which surpasses the greedy sequence in the fourth term.

This poset seems uncomfortably close to the setup we work with to compute the reducible secant indices. However, no such discrepancy arises in our computations due to the following consequence [45, Proposition 3.12] of the Clements-Lindstr¨omtheorem.

Lemma 2.4.11. Let R = {f1, . . . , fr} ⊆ k[x0, . . . , xn−1] be a regular sequence of monomials, with degrees ej = deg(fj), e1 ≤ ... ≤ er. Let N be any homogeneous ideal containing R. Then there

31 e1 er exists a lex ideal L ⊆ k[x0, . . . , xn−1] such that N and (x0 , . . . , xn−1) + L have the same Hilbert series.

By a lex ideal, we mean a monomial ideal I of k[x0, . . . , xn−1] such that the degree d piece of

I, Id, is generated by an initial segment of the lexicographic sequence of degree d monomials, for every d. This solves the quotient ring dimension problem for us because of the following.

(n) Lemma 2.4.12. Let a1, . . . , ar, 0 ≤ r ≤ Nd − n − 1 be any sequence of monomials of degree d d d in k[x0, . . . , xn] so that x0, . . . , xn−1, a1, . . . , ar are all linearly independent. Let b1, . . . , br be the initial segment of the lexicographic sequence of degree d monomials in k[x0, . . . , xn], excluding the d d d d 0 0 d d pure powers x0, . . . , xn. Then letting I = (x0, . . . , xn−1) + (a1, . . . , ar) and J = (x0, . . . , xn−1)) +

(b1, . . . , br), we have

0 0 dimk(k[x0, . . . , xn−1]/I ) ≤ dimk(k[x0, . . . , xn−1]/J ),

0 0 where I ,J denote I,J dehomogenized with respect to xn, respectively.

d d Proof. By Lemma 2.4.11, since x0, . . . , xn−1 is a regular sequence, we know that there is a lex ideal d d L such that I has the same Hilbert series as the ideal (x0, . . . , xn−1) + L. This Hilbert series is d d smaller at each term than that of the ideal (x0, . . . , xn−1) + Ld. Since in particular the degree d d d parts of the ideals I and (x0, . . . , xn−1) + L must have the same dimensions as k-vector spaces, we know that Ld is generated by the initial segment of the lexicographic sequence of degree d monomials, excluding the pure powers, of length r.

Denoting by b1, . . . , br these monomials, in lexicographic order, we arrive at the desired result.

This lemma proves that the two sequences of reducible secant indices and greedy indices are the same, and together with our previous observations in this section, shows their terms are identical to those that arise from the poset problem associated to the polynomial ring quotients above.

(n) n Theorem 2.4.13. Let X = vd (P ), n, d > 1. Then

RL(X) = RLG(X),

(n) n (n,d) (n,d) and RL (n) (vd (P )) = dimk(k[x1, . . . , xn])/Ij where the ideal Ij is the dehomoge- Nd −1−n−j d d nization with respect to xn of the sum of the ideal (x0, . . . , xn−1) with the ideal generated by the

32 (n) lexicographic sequence of degree d monomials excluding pure powers of length Nd − n − j in k[x0, . . . , xn].

It would be interesting to have a simple formula that produces these sequences. For n = 2 or for d = 2, the corresponding sequences as the other number varies follow simple patterns. But as n, d both increase these patterns become increasingly complex, as we will soon illustrate with several examples. Our expectation is that these two sequences are in fact also equal to L(X). We leave this as a conjecture.

Conjecture 2.4.14. For each n, d > 1,

RLG(X) = RL(X) = L(X),

(n) n where X = vd (P ). In other words, the maximal number of points that can be contained in the intersection of r linearly independent degree d hypersurfaces in n is RLG (X), given that the P Nd−r−1 intersection is finite and reduced.

If true, Conjecture 2.4.14 could be thought of a quasi-enumerative version of B´ezout’stheorem. There is some evidence for it from the so-called Eisenbud-Green-Harris conjecture.

Conjecture 2.4.15 (EGH [17]). Let I be a homogeneous ideal in the polynomial ring k[x0, . . . , xn−1] containing a regular sequence f1, . . . , fn of degrees deg(fi) = ai, where 2 ≤ a1 ≤ ... ≤ an. Then I

a1 an has the same Hilbert function as an ideal containing x0 , . . . , xn .

Proposition 2.4.16. The truth of the EGH-conjecture would imply that of Conjecture 2.4.14.

n Proof. Suppose we have any complete intersection of degree d hypersurfaces T1,...,Tn in P , and let Tn+1,...,Tn+r be degree d hypersurfaces so that T1,...,Tn+r are linearly independent and

T1 ∩ ... ∩ Tn+r is reduced. Write Tj = V (Fj), Fj ∈ k[x0, . . . , xn] for each j. So the F1,...,Fn form a regular sequence. Suppose the EGH-conjecture is true. Then by [12, Proposition 9] and Lemma 2.4.11 see that d d the ideal (F1,...,Fn+r) has the same Hilbert series as an ideal of the form (x0, . . . , xn−1)+L, where L is a lex ideal. The degree d part of L must then be generated by the lexicographic sequence of monomials of degree d excluding the pure powers and of length r. Thus

dimk(k[x0, . . . , xn−1]/I) ≤ dimk(k[x0, . . . , xn−1]/J),

33 d d where I,J are the ideals (F1,...,Fn+r), (x0, . . . , xn−1)+Ld dehomogenized with respect to xn.

We conclude this section with several computations of the greedy sequence of secant indices and G (n) n a specific instance of Conjecture 2.4.14. For a given n, d, computing RL (vd (P )) can be done easily using a computer algebra system such as SageMath [52].

(n) n Example 2.4.17. Let X = vd (P ). n = 2

• d = 2, RLG(X) = (1, 2, 3, 4),

• d = 3, RLG(X) = (1, 2, 3, 4, 5, 6, 7, 9),

• d = 4, RLG(X) = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16),

• d = 5, RLG(X) = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 20, 21, 25),

• d = 6,

RLG(X) =(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 24, 25, 26, 30, 31, 36),

• d = 7,

RLG(X) =(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 28, 29, 30, 31, 35, 36, 37, 42, 43, 49).

n = 3

• d = 2, RLG(X) = (1, 2, 3, 4, 5, 6, 8),

• d = 3, RLG(X) = (1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 18, 19, 21, 27),

• d = 4,

RLG(X) =(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 17, 18, 19, 20, 21, 22, 24, 25, 28, 32, 33, 34, 36, 37, 40, 48, 49, 52, 64),

n = 4

• d = 2, RLG(X) = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16),

34 • d = 3,

RLG(X) =(1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 18, 19, 21, 27, 28, 29, 30, 31, 33, 36, 37, 39, 45, 54, 55, 57, 63, 81),

n = 5

• d = 2, RLG(X) = (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32),

• d = 3,

RLG(X) =(1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 18, 19, 21, 27, 28, 29, 30, 31, 33, 36, 37, 39, 45, 54, 55, 57, 63, 81, 82, 83, 84, 85, 87, 90, 91, 93, 99, 108, 109, 111, 117, 135, 162, 163, 165, 171, 189, 243),

Note when d = 2, the gaps in these sequences are similar to those observed in [17, pg. 193].

There they consider the maximal possible dimensions of quotients k[x0, . . . , xn−1]/I where I is a homogeneous ideal generated by r linearly independent quadrics. Because lower degree polyno- mials are not allowed as generators, their sequences are shorter. This difference becomes more pronounced if one considers the same numbers for ideals generated by linearly independent degree d homogeneous polynomials rather than allowing lower degree generators as we do here. G (n) n When n = 2, the pattern determining RL (vd (P )) is straightforward and yields a particularly attractive incarnation of Conjecture 2.4.14. It is the continuation of the patterns observed in the above sequences for n = 2.

35 (2) Conjecture 2.4.18. Fix d > 0 and set N = Nd . Consider the sequence (a1, . . . , aN−3) with terms (organized first to last)

d − 1

1

d − 2

1, 1

...

4

1,..., 1, (repeated d − 4 times)

3

1,..., 1, (repeated d − 3 times)

2

1,..., 1, (repeated 3d − 3 times)

Let (b1, . . . , bN−2) be the sequence defined by

j−1 n X bj = d − ai. i=1 Then the maximal number of points that could be contained in the intersection of r linearly inde- pendent degree d curves is br−1, given that the intersection is finite and reduced.

Note it is also an implication of the EGH-conjecture, so any counterexample would also suffice to disprove the EGH-conjecture.

2.5 Segre varieties

In this section we will devise a method for computing the greedy reducible secant indices for n m N the images of Segre embeddings. Let n, m > 0 and throughout denote by σn,m : P × P ,→ P the n m N Segre embedding, where N = (n + 1)(m + 1) − 1, and let X = σn,m(P × P ) ⊆ P . The degree n+m
of X is n , so this is the top term of L(X).

36 Example 2.5.1. When m = 1 is fixed and n is allowed to vary (or vice versa), X has dimension N n + 1 inside P where N = 2(n + 1) − 1 = 2n + 1. Thus X has codimension n and its degree is n+1
N n = n + 1. Therefore X is a minimal degree subvariety of P and so by Theorem 2.2.12 its sequence of secant indices is L(X) = (1, 2, . . . , n + 1).

Therefore the question of computing the sequences of secant indices only becomes interesting 2 2 8 for n, m > 1. The first such example is P × P which is a codimension 4 subvariety of P of degree 6. So its sequence of secant indices must contain exactly one gap of size 2. In this section we will describe a method which can be used to compute RLG(X). These computations show that

G 2 2 RL (σ2,2(P × P )) = (1, 2, 3, 4, 6),

2 2 suggesting that the gap in the true sequence L(σ2,2(P × P )) occurs between the final two terms, 2 9 like for the example of the degree 3 Veronese embedding of P into P . One can verify that in fact

2 2 L(σ2,2(P × P )) = (1, 2, 3, 4, 6).

The example is simple enough that we can conclude by Theorem 2.2.12 and Proposition 2.3.9, as 2 2 σ2,2(P × P ) is smooth and nondegenerate, and is not a rational scroll.

Proposition 2.5.2. 2 2 L(σ2,2(P × P )) = (1, 2, 3, 4, 6).

(2) 2 Proposition 2.3.9 could also be used to prove L(v3 (P )) = (1, 2, 3, 4, 5, 6, 7, 9) in place of the classical Cayley-Bacharach theorem. In this sense, this classification result, and results such as that of Noma [48], fulfill similar roles to the Cayley-Bacharach theorem but in the case of arbi- trary smooth nondegenerate projective subvarieties. However, they only concern extremal secant subspaces and do not provide lower bounds for the secant indices, so do give us a means to answer whether L(X) = RLG(X) for the more complicated instances of Segre and Veronese varieties.

37 Because of this, we focus here on computing instead RLG(X) to obtain conjectural values for n m n L(X), for X = σn,m(P × P ). To derive this method, note that if we choose coordinates, P (x0 : m N ... : xn), P (y0 : ... : ym), P (z0 : ... : zN ) then for any hyperplane H = V (a0z0 + ... + aN zN ) of N P , we see that X ∩ H is the zero locus of the polynomial a0x0y0 + a1x0y1 + ... + aN xnym. This quadric can split into at most two factors. On the other hand, we can choose any collection of n + m hyperplanes

n H1,...,Hn+m ⊆ P with the property that any subset of n of these hyperplanes has only a single point in common, and any n + 1 do not have any point in common. Likewise we can choose an analogous collection 0 0 m of hyperplanes H1,...,Hn+m ⊆ P with the property that any subset of m of those hyperplanes meet at a single point and no m + 1 have a point in common. Each subscheme

0 n m Tj := Hj ∪ Hj ⊆ P × P

n m is then a reducible and reduced hyperplane section of P × P via the Segre embedding, and we Tn+m n+m
n m have j=1 Tj is finite and reduced, realizing the degree n of P ×P in cardinality. Any n+m N hyperplanes of P meeting at a dimension N − n − m-dimension linear subvariety that meets X in a finite and reduced collection of points will be of this form. As in Section 2.4, there is symmetry here in the sense that the points of the intersection

T1 ∩ ... ∩ Tn+m are exactly those of the form

0 0 (Ha1 ∩ ... ∩ Han ) × (Hb1 ∩ ... ∩ Hbm ) where {1, 2, . . . , n + m} = {a1, . . . , an} ∪ {b1, . . . , bm}. Therefore, we can strip the essence of the computation of RLG(X) from the context of the Segre embedding and find it equivalent to the following problem.

Combinatorial problem 2.5.3. Let Y be the set of all tuples (A, B) where A is a set of size n and B a set of size m so that A ∪ B = {1, 2, . . . , n + m}. Let S be the set of all sequences of length nm consisting of distinct tuples (a, b) where a, b ∈ {1, 2, . . . , n + m} and a 6= b. Given a tuple (a, b), we say we are cutting Y by (a, b) if we replace Y with the subset of elements (A, B) of

Y for which either a ∈ A or b ∈ B. For each sequence ((a1, b1),..., (anm, bnm)), form an integer

38 sequence (c0, . . . , cnm) where cj is the cardinality of the set obtained by cutting Y by each (ai, bi) for i = 1, . . . , nm − j. Note cnm = |Y |. The problem is then to compute the maximal (with respect to the total order ≤ of Section 2.3) possible integer sequence arising in this manner that is also strictly increasing.

The combinatorial problem gives us a clearer way to compute the reducibility of X and find µ(X).

n m Proposition 2.5.4. Let X = σn,m(P ×P ). Then the reducibility of X is 2, and µ(X) = 2 dim(X).

Proof. Because of our observations above, we see that the most a hyperplane section of X can split is into two components, so the reducibility of X is at most 2. All that must be done is exhibit a sequence of hyperplane sections satisfying the conditions of 2-reducibility that also has the minimum total number of irreducible components. To help simplify the notation, we use the notation of Combinatorial problem 2.5.3. Let p =

(A, B) be any point of Y , and write A = {a1, . . . , an}, B = {b1, . . . , bm}. Then consider the set T consisting of the nm points obtained from swapping one element of A with one element of B.

Use pij to denote the point where ai was swapped with bj. For each such i, j, cutting by the tuple

Hi,j = (ai, bj) removes pij from Y but does not remove any of the other elements of T .

Each Hi,j corresponds to a hyperplane section of X, and the sequence

H1,1,H2,1,...,Hn,m, taken in any order, is a sequence of hyperplane sections satisfying the conditions of 2-reducibility, with 2 dim(X) total irreducible components.

Thus altogether we have the following.

Theorem 2.5.5. For a given n, m > 1, the answer to Combinatorial problem 2.5.3 is the sequence RLG(X).

It is straightforward to write an algorithm that solves this problem for a given n, m > 1. We have implemented such an algorithm using the SageMath computer algebra system [52], and have used it to compute the greedy sequence of reducible secant indices for several values of n, m. It

39 would be interesting to know if there is an analog of Lemma 2.4.11 that would work in this context to show that RLG(X) = RL(X).

Verifying this without additional theoretical support requires a brute-force check of every possible sequence of elements of the form (A, B) using the notation of Combinatorial problem 2.5.3, and this becomes impractical for even small n, m.

Example 2.5.6. m = n • n = 2, m = 2, RLG(X) = (1, 2, 3, 4, 6),

• n = 3, m = 3, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 14, 20),

• n = 4, m = 4, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 14, 20, 21, 25, 35, 36, 40, 50, 70),

m = n − 1 • n = 3, m = 2, RLG(X) = (1, 2, 3, 4, 6, 7, 10),

• n = 4, m = 3, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 14, 20, 21, 25, 35),

m = n − 2 • n = 4, m = 2, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 15),

• n = 5, m = 3, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 14, 20, 21, 25, 35, 36, 41, 56),

m = n − 3 • n = 5, m = 2, RLG(X) = (1, 2, 3, 4, 6, 7, 10, 11, 15, 16, 21).

It seems reasonable to expect that for this problem, like for that of the Veronese varieties, the greedy sequence is equal to the sequence of secant indices. We leave this as a conjecture.

n m Conjecture 2.5.7. For any n, m > 1, X = σn,m(P × P ), we have

RLG(X) = L(X).

While it may be difficult to write down a closed form formula for RLG(X), one simple compu- tation is determining the size of the last gap in RLG(X). A special case of the above conjecture is then that the same gap must be present in the sequence of secant indices.

Conjecture 2.5.8. n + m − 2 L (X) − L (X) = . nm nm−1 n − 1

40 2.6 Lines on surfaces

Our original motivation comes from the classical problem of determining the maximal number 3 m`(d) of lines that can be contained in a degree d smooth surface S of P (x : y : z : w). As all surfaces of degree d ≤ 2 are ruled, this question is only relevant for surfaces of degree d ≥ 3. When d = 3, the Cayley-Salmon theorem ensures that every such S must contain exactly 27 distinct lines. However, for d ≥ 4, the general degree d surface contains no lines at all; the problem of determining m`(d) is a so-called quasi-enumerative problem [20].

Over a century ago, Clebsch [13] formulated the bound m`(d) ≤ d(11d − 24) and Segre [51] later proved m`(d) ≤ (d − 2)(11d − 6), for all d ≥ 3. These bounds have been improved slightly in modern times and the methods of Segre made rigorous using modern intersection theory. Bauer and Rams [9, Theorem 1.1] have proven

2 m`(d) ≤ 11d − 32d + 24 for all d ≥ 3. To our knowledge, this is the best known bound for m`(d) when d ≥ 6. However, this latest bound is still known not to be sharp. It fails to be so already for the case of smooth quartic surfaces, where it is known that any irreducible quartic surface not ruled by lines contains at most 64 lines, see for instance [26, Theorem 4.5]. Furthermore, quartics achieving this bound exist. To find a lower bound for m`(d) one only needs to provide an example of a surface with lines. For general d, the best known example of a surface with many lines is the smooth degree d Fermat surface V (xd + yd + zd + wd), which contains exactly 3d2 distinct lines.

The first of the numbers m`(d) which is currently unknown is that for d = 5. By [50, Theorem

1.2], together with the example of the Fermat quintic, we see that 75 ≤ m`(5) ≤ 127. To our knowledge, no example is currently known of a smooth quintic surface with more than 75 lines.

Interestingly, because m`(d) is bounded between two quadratic polynomials, if it turns out that m`(5) ≤ 101, then by interpolation using the examples m`(3) = 27, m`(4) = 64, there can be no polynomial function f(d) that agrees with m`(d) for all d ≥ 3. If this were true, then it would imply that any method which can only produce polynomial bounds is doomed to fail to produce an exact formula for m`(d). Our work presented in this chapter began by looking for an alternate description of the numbers m`(d). In what follows, we relax the constraint on the surfaces we consider to allow for nonreduced,

41 reducible, and singular surfaces. Let m`(d) now denote the maximal possible number of lines that 3 can be contained in a degree d surface of P , given that the surface contains only finitely many lines. There is a generalization of the Veronese embedding for Grassmannians [30]; in particular, given a d > 1, one may define an embedding

d + 3 d + 3 v : (1, 3) = G(2, 4) ,→ G − d − 1, d G d d by

L 7→ I(L)d,

3 for each line L ⊆ P , where I(L)d denotes the degree d part of the homogeneous ideal I(L) of L. Given a nonzero homogeneous polynomial F ∈ k[x, y, z, w] of degree d, one can consider the d+3
d+3
d+3
subset of G( d − d − 1, d ) consisting of all d − d − 1-planes of k[x, y, z, w]d containing d+3
d+3
F . This is a special type of Schubert subvariety of G( d − d − 1, d ), a sub-Grassmannian, d+3
d+3
isomorphic to G( d − d − 2, d − 1). Then note that we can express m`(d) as the maximal

finite intersection that can occur by intersecting vd(G(1, 3)) such sub-Grassmannians,

m`(d) = max{|vd(G(1, 3)) ∩ G| | |vd(G(1, 3))| < ∞},

d+3
d+3
where this maximum is taken over all sub-Grassmannians G in G( d − d − 1, d ) of the above form. All this generalized Veronese embedding serves to do is provide us with an alternate language with which to state our problem. However, when viewed from this perspective, it suggests ways to relate m`(d) to several integer sequences that seem interesting in their own right. A first naive sequence works as follows.

Definition 2.6.1. Let X ⊆ G(k, n) = G(k+1, n+1) be any subvariety. For each i = 0,..., codim(X), where the codimension of X is calculated as a subvariety of G(k, n), define

Li(X) := max{|X ∩ T | | |X ∩ T | < ∞}, where this maximum is taken over all Schubert subvarieties T of G(k, n) of dimension i.

42 The reason for the abuse of notation in redefining L(X) here is that when G(k, n) is a projective space, such as when n = 0, this definition indeed specializes to our first definition of the secant indices described in Section 2.2, without the condition that X be smooth or that these intersections be reduced. Note that the top term of L(X) is deg(X), where this degree is that of the image of X by the

Pl¨ucker embedding of G(k, n) in the codomain projective space. In the specific case of interest, when X = vd(G(1, 3)) and G(k, n) the codomain of vd, we have

m`(d) ≤ Lt(X), where d + 3  t = − d − 2 (d + 1) d d+3
d+3
is the dimension of any sub-Grassmannian of the form G( d −d−2, d −1). Thus deg(vd(G(1, 3))), which may be computed using basic intersection theory, provides an upper bound for m`(d), but is far too large to be of any practical use. Nevertheless, it seems interesting to ask about the sequences

L(X) for arbitrary subvarieties X ⊆ G(k, n).

When X = vd(G(1, 3)) one can define an alternate and shorter sequence by instead only considering intersections of X with certain sub-Grassmannians. Denote by GV the subset of d+3
d+3
G( d − d − 1, d ) of subspaces in k[x, y, z, w]d containing a fixed subspace V . For each 1 ≤ i ≤ k, we can define the number

max{|X ∩ GV | | dimk(V ) = r, |X ∩ GV | < ∞}.

The bottom term of the resulting integer sequence, when i = 1, is exactly m`(d), and as GV is a single point when V is k-dimensional, the top term of the sequence is just 1 . Described differently, the question of what the terms of this sequence are is equivalent to the following.

Question 2.6.2. What is the maximal number of lines that can be contained in the intersection d+3
of 1 ≤ i ≤ d − d − 1 linearly independent degree d surfaces (possibly singular, reducible, or 3 nonreduced) of P , supposing that the intersection contains only finitely many lines?

d+3
The fact that the intersection of any d − d − 1 degree d surfaces can only contain 1 line is d+3
3 a consequence of the fact that dimk(I(L)d) = d − d − 1 for any line L ⊆ P . One could further

43 modify the question to only allow the intersection of smooth degree d surfaces. In this case, it is straightforward to see that the terms of the resulting integer sequence are nondecreasing as one d+3
varies i from d −d−1 to 1. The last term of this sequence is the m`(d) defined at the beginning of this section for smooth surfaces, and the penultimate term is at most d2 by B´ezout’stheorem. d+3
As an example, for d = 4, this sequence has d − d − 1 = 30 terms, but we already know the penultimate term is at most 16 and the first term is 1. So the sequence must contain repeated numbers. Our original hope was that if one could compute the earlier numbers of this sequence, then it would be possible to extrapolate from observable patterns in those numbers a formula for m`(d). The feasibility of this approach appears questionable now, but these sequences seem to be of independent interest, potentially reflecting properties of the possible configurations of lines on degree d surfaces.

44 CHAPTER 3

LINEAR RECURRENCE SEQUENCES AND THE DUALITY DEFECT CONJECTURE

3.1 Introduction

This chapter concerns the duality defect conjecture, which may have first been posed by Alan Landman in 1974:

N 2N ∨ Conjecture. Let X be a smooth nonlinear subvariety of P with dim(X) > 3 . Then X is a hypersurface, or equivalently def(X) = 0.

∨ N∨ Here X denotes the dual variety of X, the subvariety of dual projective space P consisting of all hyperplanes tangent to X, and def(X) = N − 1 − dim(X∨) is called the duality defect of X, which provides a measure of how far X∨ is from being a hypersurface. Our aim is to give a generalization of an algorithm of Oaland [49] for proving the conjecture in the codimension 3 case to the higher codimension cases of the conjecture and to prove the following two results:

N Theorem 3.4.8. If X is a smooth nonlinear codimension m subvariety of P with duality defect 3N−2 r > 0 and N − m ≥ 4 , then

m j X N − m − r  deg(X) ≤ . 2 j=0

N Theorem 3.5.1. The duality defect conjecture in the codimension 3 case is true for P when N is odd.

Some of the motivation for this work comes from a well-known conjecture of Hartshorne [31] on complete intersections.

N 2N Conjecture. If X is a smooth subvariety of P with dim(X) > 3 , then X is a complete inter- section.

45 This conjecture still remains open today, over four decades after its declaration, even for the case of subvarieties of codimension 2. Its truth would imply that of the duality defect conjecture, see for instance [15, Proposition 3.1] or [36, Proposition 2.1]. So far the duality defect conjecture has proven to be a more tractable problem; it is known to be true in the codimension 2 case, with proofs given by Ein [15, Theorem 3.4] and independently by Holme and Schneider [37, Corollary 6.4]. A proof of the duality defect conjecture would provide evidence for Hartshorne’s conjecture, and any counterexample would also serve to disprove Hartshorne’s conjecture. We are interested in a combinatorial approach to proving the duality defect conjecture that Holme developed which was used to give an alternate proof of the codimension 2 case [36]. This approach is promising as it may be extendable to the higher codimension cases. Oaland [49] used Holme’s approach and results of Ein [15, Theorem 2.3] and Zak [25, Corollary 7.4] to create an algorithm for proving the conjecture in the codimension 3 case for individual values of N, and was able to use this to prove the conjecture in that case for N = 10, 11,..., 141. N Specifically, given a smooth subvariety i : X,→ P of dimension n the combinatorial approach ∨ N N∨ works by considering the class of the conormal variety Z(X) = P (NX/PN (−1) ) ,→ P × P in N N∨ the Chow ring A(P × P ). This class may be written as

N N−n n+1 [Z(X)] = δ0(X)s t + ... + δn(X)s t ,

N N∨ N N N∨ N∨ for some δj(X) ∈ Z, where if pr1 : P × P → P and pr2 : P × P → P are the projection ∗ ∗ 0 N 0 N∨ maps, s = pr1([H]) and t = pr2([H ]) for hyperplanes H ⊆ P , H ⊆ P . It is shown by

Holme [36] that for each j, we have δj(X) ≥ 0, and, if δj(X) = 0 for j = 0, . . . , r −1, and δr(X) 6= 0 for some r ∈ {0, . . . , n}, then def(X) = r. Thus the duality defect of X can be computed from the coefficients of the class of Z(X). In particular, the dual variety of X is a hypersurface if and only if δ0(X) 6= 0. ∨ ∨ Additionally, each δj(X) is in fact equal to deg(sn−j(NX/PN (−1) )), where sj(NX/PN (−1) ) ∨ denotes the jth Segre class of the bundle NX/PN (−1) using the conventions of Fulton [23]. By applying properties of singular cohomology for complex projective varieties one may show that 3N−2 N−n when n ≥ 4 , for our purposes we may assume c(NX/PN (−1)) = 1 + c1h + ... + cN−nh for ∗ some cj ∈ Z, where c(NX/PN (−1)) is the total Chern class of NX/PN (−1) and h = i ([H]). This

46 allows us to write

∨ 1 n s(NX/PN (−1) ) = ∨ = 1 + s1h + ... + snh c(NX/PN (−1) ) for some sj ∈ Z, and where each δj(X) = deg(X)sn−j. These sj form part of a homogeneous integer linear recurrence sequence of order N − n:

s0 = 1, j X q+1 sj = (−1) cqsj−q, for j = 1,...,N − n − 1 q=1 (3.1) N−n X q+1 sj = (−1) cqsj−q, for j ≥ N − n. q=1 The algorithm of Oaland [49, Kapittel 5] works in the codimension 3 case by showing that if

X has positive defect then there are only finitely many possibilities for c1, c2, c3, and the possible sequences of sj can be computed. One of the ideas of Holme [36] applied in Oaland’s work is to use positivity properties that the Schur polynomials in the Chern classes of NX/PN (−1) satisfy to obtain inequalities involving the cj. Using this idea, we find inequalities that extend the algorithm of Oaland to the higher codimension cases of the conjecture. These inequalities also follow from a recent result of Huh [38, Theorem 21] which specifies additional constraints the coefficients of N N∨ the class of a subvariety in A(P × P ) must satisfy, and are key to our proof of Theorem 3.4.8. By running the generalized algorithm we show that the duality defect conjecture is true in the codimension 3 case for N = 10, 11,..., 201, in the codimension 4 case for N = 14, 15,..., 50, and in the codimension 5 case for N = 18, 19 ..., 23.

Alternatively, if one can show there are simply no recurrence sequences of this form with sn =

... = sn−r+1 = 0 and sj > 0 for j ≤ n − r for appropriate r > 0, then there can be no positive N defect smooth nonlinear subvarieties of P of dimension n. In this way, a better understanding of the positivity of homogeneous integer linear recurrence sequences can have geometric implications. This is the idea of our proof of Theorem 3.5.1, where we reduce the necessary work to proving the following purely number-theoretic statement.

Lemma 3.5.2. Let (uj)j∈Z≥0 be the homogeneous integer linear recurrence sequence defined by the initial conditions u0 = 0, u1 = 0, u2 = 1, and uj = c1uj−1 − c2uj−2 + c3uj−3 for j > 2, where c1, c2, c3 ∈ Z>0. Suppose there exists an m ∈ Z>2 with um = um+1 = 0 and uj > 0 for j = 2, . . . , m − 1. Then m = 4 or 6.

47 This chapter is organized as follows. In Section 3.2 we introduce conventions, notations, and facts that will be used throughout. In Section 3.3 we define the needed concepts from projective duality and describe how the duality defect of a smooth projective subvariety may be computed from the coefficients of the class of its conormal variety. In Section 3.4 we highlight the connection between the duality defect conjecture and homogeneous integer linear recurrence sequences and then give a generalization of the algorithm of Oaland to the higher codimension cases of the conjecture. Here we also prove Theorem 3.4.8. Lastly, Section 3.5 is dedicated to the proof of Theorem 3.5.1. Note Thanks to Mark van Hoeij for pointing out to me that Lemma 3.5.2 could be sharpened to its current form. The work in this chapter was partially supported by NSA grant H98230-16-1-0016.

3.2 Preliminaries

Throughout we take our base field to be C. For an algebraic vector bundle E of rank e on a vari- ety X of dimension n, we define its projective bundle to be the X-scheme P (E) = Proj(Sym (E∨)) OX where E is the locally free sheaf of sections of E, following the conventions in Fulton [23]. Our definition is contrary to the convention used by Holme [36] and Grothendieck, where our P (E) ∨ ∨ ∨ would be denoted P(E ). Here E denotes the dual of E, that is, the sheaf E = HomOX (E, OX ). ∨ We will also denote the dual bundle of E by E and by E(d) the bundle E ⊗ OX (d). Additionally, we use Fulton’s conventions for Chern and Segre classes [23, Chapter 3]. To simplify notation, we will denote the jth Segre class of E by sj(E) = sj(E) ∩ [X] which lives in the Chow group An−j(X), and by cj(E) = cj(E) ∩ [X] ∈ An−j(X) the jth Chern class of E.

The total Chern class of E will be denoted c(E) = 1 + c1(E) + ... + ce(E), and the Chern and P∞ j Segre polynomials in indeterminant t of E will be denoted by ct(E) = j=0 cj(E)t , st(E) = P∞ j j=0 sj(E)t , respectively. The Chern and Segre classes of E are related via 1 st(E) = ct(E) and so:

Pj Lemma 3.2.1. We have s0(E) = 1, and sj(E) = − q=1 cq(E)sj−q(E) for every j ∈ Z>0.

48 Suppose λ = (λ1, . . . , λm) represents a partition of an integer m ∈ Z>0 into integers e ≥ λ1 ≥

... ≥ λm ≥ 0. We define the Schur polynomial corresponding to λ in the Chern classes of E to be

∆λ(E) = det(cλi+j−i(E))1≤i,j≤m   cλ1 (E) cλ1+1(E) ··· cλ1+m−1(E)

 cλ2−1(E) cλ2 (E) ··· cλ2+m−2(E) = det   ,  ············ 

cλm−m+1(E) cλm−m+2(E) ··· cλm (E) adopting the notation cj(E) = 0 for j < 0. From Fulton [23, Example 12.1.7] we have:

≥ ≥ Theorem 3.2.2. If E is globally generated, then ∆λ(E) ∈ An−m(X) where we denote by An−m(X) the subset of An−m(X) consisting of the classes that can be represented by nonnegative cycles.

One of the key ingredients to our approach is a topological result due to Larsen [44] concerning the singular cohomology of complex projective varieties.

N Theorem 3.2.3. Let i : X,→ P be a smooth subvariety of codimension r. Then the maps j N j H (P , Z) → H (X, Z) induced by i are isomorphisms for j ≤ N − 2r.

We abuse notation, using the same symbols to denote both varieties in the algebraic context and their analytifications. Connecting the Chow and singular theories, there are cycle maps Aj(X) →

H2n−2j(X) inducing a group homomorphism cl∗ : A(X) → H∗(X) covariant for proper morphisms, and after composition with the isomorphisms from Poincar´eduality, a ring homomorphism cl∗ : ∗ N A(X) → H (X, Z) contravariant for morphisms of smooth varieties. Since P admits a cellular 1 2 N N N decomposition ∅ ⊆ P ⊆ P ⊆ ... ⊆ P , the cycle map cl∗ : A(P ) → H∗(P ) is an isomorphism. A reference for these facts is Fulton [23, Chapter 19]. j If α ∈ A (X), we define the degree of α to be deg(α) = a, with a ∈ Z such that i∗(α) = N−n+j N N a[H] ∈ A(P ), where [H] is the class of a hyperplane in P . Similarly, for an element α ∈ ∼ N+1 Hj(X), we define deg(α) ∈ Z using the isomorphism H∗(X) = Z[s]/(s ). Since the cycle map cl∗ is covariant for proper morphisms we have that if α ∈ Aj(X) for some j, then deg(α) = deg(cl∗(α)).

3.3 Dual varieties and duality defect

N∨ We denote dual projective space of dimension N by P . This is projective space of dimension N N, but with its C-points identified with the hyperplanes of P . In the classical language:

N∨ N P = {H ⊆ P | H is a hyperplane}.

49 N N N∨ If we choose coordinates P (x0 : ... : xN ), then the natural identification of P with P is N defined so that (a0 : ... : aN ) corresponds to the hyperplane of P cut out by a0x0 + ... + aN xN ∈ N C[x0, . . . , xN ]. Suppose i : X,→ P is a subvariety of dimension n, where i denotes the inclusion map, and let (F1,...,Fr) be its homogeneous ideal. For p ∈ X, we define the embedded tangent N space of X at p, denoted TX,p, to be the linear subvariety of P cut out by the polynomials

∂Fj ∂Fj (p)x0 + ... + (p)xN . ∂x0 ∂xN

We define the conormal variety of X to be

◦ N N∨ Z(X) = {(p, H) | TX,p ⊆ H, p ∈ X } ⊆ P × P , where X◦ denotes the smooth locus of X. From now on we assume X is smooth, in which case N N∨ N taking the closure of the set in the definition is unnecessary. Let pr1 : P × P → P , pr2 : N N∨ N∨ P × P → P be the projection maps. The map pr1 induces a map pr1 : Z(X) → X the fibers of which are irreducible of dimension N −n−1 and in fact identifies Z(X) with a projective bundle ∗ over X. Specifically, the restriction of the line bundle pr2(OPN∨ (−1)) to Z(X) is a subbundle of ∗ ∨ ∼ ∨ pr1(NX/PN (−1) ) which induces an isomorphism Z(X) = P (NX/PN (−1) ) compatible with the projection maps [23, Appendix B, 5.5]. See also [36, Section 1], though note the differences there in projective bundle conventions. As an immediate consequence of this identification we have dim(Z(X)) = N − 1. N∨ ∨ We define the dual variety of X in P , denoted by X , to be the image of Z(X) by pr2. Thus ∨ 2N dim(X ) ≤ N − 1. The duality defect conjecture claims that when X is nonlinear and n > 3 , this inequality is actually equality, or in other words the duality defect of X, def(X) = N −1−dim(X∨), is zero. To study Z(X) and X∨, we first give a more useful description of the embedding of Z(X) into N N∨ P × P for our purposes. We will do this by showing that NX/PN (−1) is globally generated, N+1 and will furnish a surjective morphism OX → NX/PN (−1). The following approach is that of Holme [36], [35] and Oaland [49], but we describe it here in slightly greater generality for later use. N+1 Suppose G is a globally generated vector bundle on X with surjection OX → G. Then this map induces a closed embedding

P (G∨) ,→ P ((ON+1)∨) = Proj(Sym (ON+1)) = X × N . X OX X P

50 After composition with the maps

N N N ∼ N N∨ X × P ,→ P × P = P × P ,

∨ N N∨ ∨ this gives an embedding P (G ) ,→ P × P , enabling us to consider the class of P (G ) in the N N∨ N N∨ Chow ring A(P ×P ). Let pr1, pr2 be the projection maps from P ×P like before. Note that N N N∨ N∨ O N+1 ∨ (1) is the pullback of O N∨ (1) by the composite map X × P ,→ P × P → P . P ((OX ) ) P N N∨ ∼ N+1 N+1 ∗ There is an isomorphism A(P × P ) = Z[s, t]/(s , t ) identifying s, t with pr1([H]), ∗ 0 N 0 N∨ ∨ pr2([H ]) respectively, for hyperplanes H ⊆ P , H ⊆ P . If P (G ) has dimension d, then ∨ N N∨ [P (G )] is a homogeneous element of A(P × P ) of degree r = 2N − d with respect to the natural grading by codimension, and so there exist a0, . . . , ar ∈ Z such that

∨ r r−1 r−1 r [P (G )] = a0s + a1s t + ... + ar−1st + art . (3.2)

∨ These aj can be computed from the Segre classes of G . This follows from Scott’s formula [23, pg. 61]:

Theorem 3.3.1. Let E be a vector bundle on X, and let F be a subbundle of E, with quotient bundle G. Suppose G has rank q. There is a canonical closed embedding P (F ) ,→ P (E), which allows us to consider [P (F )] as a class in A(P (E)). Then q X j ∗ [P (F )] = c1(OP (E)(1)) p (cq−j(G)) ∈ A(P (E)), j=0 where p : P (E) → X is the structure morphism.

Applying this formula to the exact sequence

∨ N+1 ∨ ∨ 0 → G → (OX ) → F → 0,

N+1 where F is the kernel of the surjection OX → G, and arguing analogously to Holme [36, Section 1] establishes:

Lemma 3.3.2. We have q ∨ X j ∗ ∨ N [P (G )] = t p (sq−j(G )) ∈ A(X × P ), j=0

∗ 0 N where q is the rank of F , and t denotes the pullback of t = pr2([H ]) by the inclusion X × P ,→ N N∨ ∨ P × P . Furthermore, aj = deg(sq−j(G )) for each j = 0, . . . , q.

51 The normal bundle NX/PN is defined from the exact sequence

∗ 0 → TX → i (TPN ) → NX/PN → 0.

Tensoring by a line bundle preserves exactness, so we may twist to obtain

∗ 0 → TX (−1) → i (TPN )(−1) → NX/PN (−1) → 0.

∗ N Next applying the pullback functor i (−) to the Euler sequence for P [32, II, Example 8.20.1] and then twisting yields the exact sequence

N+1 ∗ 0 → OX (−1) → OX → i (TPN )(−1) → 0.

∗ By identifying the term i (TPN )(−1) in both of these sequences we obtain the desired surjection N+1 OX → NX/PN (−1).

Replacing G with NX/PN (−1) in equation (3.2) yields the simpler expression

∨ N N [Z(X)] = [P (NX/PN (−1) )] = a1s t + ... + aN st .

When referring to the class of Z(X), we adopt the notation aj+1 = δj(X), and refer to the δj(X) as the delta invariants or the degrees of the polar classes of X, see [35, Section 3]. It follows from the N−n−1 n+2 N projection formula that in fact δn+1(X)s t + ... + δN−1(X)st = 0, and so the expression for the class of the conormal variety simplifies further to

N N−n n+1 [Z(X)] = δ0(X)s t + ... + δn(X)s t .

From Holme [36, Theorem 1.1], we are able to read off the duality defect of X from the δj(X):

∨ Theorem 3.3.3. Let r ∈ {0, . . . , n}. If δ0(X) = ... = δr−1(X) = 0 and δr(X) 6= 0, then X has dimension N − 1 − r. Thus in this case, def(X) = r.

Remark 3.3.4. The argument used to prove this establishes that the δj(X) represent intersection numbers of projective subvarieties with linear subvarieties in general position, and thus that δj(X) ≥ 0 for every j. This same argument will work for any globally generated vector bundle G on X, and ∨ shows that in the expression (3.2) for the class [P (G )], each aj ≥ 0. By Lemma 3.3.2, this means ∨ that deg(sq−j(G )) ≥ 0 for j = 0, . . . , q, where q is as in Lemma 3.3.2.

52 N+1 Applying Lemma 3.3.2 to the surjection OX → NX/PN (−1) gives us δj(X) = deg(sn−j(NX/PN (−1))) for each j. From the exact sequence

N+1 0 → F → OX → NX/PN (−1) → 0,

N+1 where F denotes the kernel of the surjection OX → NX/PN (−1), we obtain a surjection

N+1 ∼ N+1 ∨ ∨ OX = (OX ) → F .

∨ This shows F is globally generated, and therefore by Lemma 3.3.2, the integers deg(sj(F )) are N N∨ the coefficients of the class of P (F ) in A(P × P ). By the sum formula for Chern classes,

N+1 ct(F )ct(NX/PN (−1)) = ct(OX ) = 1, which implies ct(NX/PN (−1)) = st(F ). Thus in particular,

deg(cj(NX/PN (−1))) = deg(sj(F )) for j = 0,...,N − n. This in addition to Remark 3.3.4 gives:

Lemma 3.3.5. For every j = 0, . . . , n,

∨ δj(X) = deg(sn−j(NX/PN (−1) )) ≥ 0, and for j = 0,...,N − n, we have

deg(cj(NX/PN (−1))) ≥ 0.

From Holme [35, Theorem 5.1] in fact:

Theorem 3.3.6. For every j = def(X), . . . , n,

∨ δj(X) = deg(sn−j(NX/PN (−1) )) > 0.

Lastly, for our purposes Theorem 3.2.3 will allow us to assume

N−n c(NX/PN (−1)) = 1 + c1h + ... + cN−nh

∗ for some cj ∈ Z and h = i ([H]), provided the codimension of X is sufficiently small. We use the following Lemma to extract precisely what we need from Theorem 3.2.3 and translate it to the algebraic context for convenience.

53 N 3N−2 Lemma 3.3.7. Let i : X,→ P be a smooth subvariety of dimension n ≥ 4 . Then for each j = 1,...,N − n, there exists cj ∈ Z such that for any j1, . . . , jN−n ∈ Z≥0 and any homogeneous α ∈ A(X), N−n N−n Y jq Y q jq deg(α cq(NX/PN ) ) = deg(α (cqh ) ). q=1 q=1 That is, for the purpose of computing the degrees of such monomials, we may assume that each j cj(NX/PN ) = cjh . The analogous result holds for the bundle NX/PN (−1).

∗ N−n Proof. By the self-intersection formula, in fact cN−n(NX/PN ) = i (i∗([X])) = deg(X)h . So we need only address the cases of cj(NX/PN ) for each j = 1,...,N − n − 1. For such a j we have the commutative diagram

j N i∗ j A (P ) A (X)

cl∗ cl∗

2j N i∗ 2j H (P , Z) H (X, Z) By Theorem 3.2.3 the bottom map of this diagram is an isomorphism, and so too is cl∗ : j N 2j N j ∗ A (P ) → H (P , Z). Thus there exists a cj ∈ Z such that cj(NX/PN ) − cjh ∈ ker(cl ). The desired result then follows from the observation that for homogeneous α, β, γ ∈ A(X), with α, β in ∗ q the same homogeneous component, and for q ∈ Z≥1, if α − β ∈ ker(cl ) = ker(cl∗) then deg(γα ) = q deg(γβ ). That the analogous result holds for NX/PN (−1) is a consequence of the formula for the Chern classes of a bundle after taking a tensor product with a line bundle [23, Example 3.2.2].

3.4 A generalized algorithm and bounds for the degrees of counterexamples

Here we will show how the facts presented in the preceding section connect the duality defect conjecture to the theory of linear recurrence sequences and we will give a generalization of the algorithm of Oaland [49] to the higher codimension cases of the conjecture. We then prove Theorem 3.4.8, stated in the introduction, which bounds the degrees of possible counterexamples to the conjecture that satisfy the constraints of Lemma 3.3.7. N 3N−2 Suppose i : X,→ P is a smooth subvariety of dimension n ≥ 4 . By Lemma 3.3.7 for our purposes we may assume there exist cj ∈ Z such that

N−n c(NX/PN (−1)) = 1 + c1h + ... + cN−nh ,

54 ∗ N where h = i ([H]) for any hyperplane H ⊆ P . Applying the formula for the Chern class of the dual of a bundle [23, Remark 3.2.3] gives

∨ N−n N−n c(NX/PN (−1) ) = 1 − c1h + ... + (−1) cN−nh .

So now, by the formula of Lemma 3.2.1, we have

j ∨ X q+1 q ∨ sj(NX/PN (−1) ) = (−1) cqh sj−q(NX/PN (−1) ) q=1 for every j ∈ Z>0. Thus there exist sj ∈ Z such that

∨ n s(NX/PN (−1) ) = 1 + s1h + ... + snh .

∨ Furthermore, since the Chern classes of NX/PN (−1) vanish beyond its rank N − n, the sj form part of the recurrence sequence (3.1):

s0 = 1, j X q+1 sj = (−1) cqsj−q, for j = 1,...,N − n − 1 q=1 N−n X q+1 sj = (−1) cqsj−q, for j ≥ N − n. q=1 ∨ By the projection formula, for each j, δj(X) = deg(sn−j(NX/PN (−1) )) = deg(X)sn−j. So in particular, the positivity of the delta invariants is the same as that of the sn−j. From Lemma

3.3.5, this implies that the sn−j are nonnegative, and additionally by Theorem 3.3.3, that for j = 0,..., def(X) − 1, sn−j = 0, and for j = def(X), . . . , n, that sn−j > 0 by Theorem 3.3.6.

Lemma 3.3.5 also shows that each cj ≥ 0. N 3N−2 This means every smooth subvariety i : X,→ P of dimension n ≥ 4 gives rise to cj ∈

Z≥0 such that the corresponding linear recurrence sequence (3.1) satisfies the above positivity constraints. If we can show that no such linear recurrence sequences exist for positive values of def(X), then there can be no positive defect subvarieties. Ein [15, Theorem 2.4] has proven the following parity result concerning duality defect:

N Theorem 3.4.1. If X is a smooth nonlinear subvariety of P with positive duality defect def(X), then def(X) ≡ dim(X) mod 2.

55 N For a general point x ∈ X and a general tangent hyperplane H in P of X at x, we define the contact locus of H with X to be the singular locus of the intersection of X and H,(X ∩ H)Sing. The above result is an immediate corollary to Ein’s more general result [15, Theorem 2.3]:

N Theorem 3.4.2. Suppose X is a smooth nonlinear subvariety of P of dimension n with positive duality defect. Then if x and H are as above, the contact locus L = (X∩H)Sing is a linear subvariety 1 of dimension def(X). Let T be a line in L, which we identify with P . Then

n−def(X) n−def(X) N =∼ O 2 ⊕ O 2 (1). L/X T P1 P1

N−def(X) Since L has dimension def(X), we also have N N =∼ O (1). These normal bundles L/P T P1 fit into the following exact sequence, see [23, Appendix B, 7.4]:

0 → NL/X T → NL/PN T → NX/PN T → 0.

It then follows from the sum formula for Chern classes that

2N − n − def(X) deg(c1(N N )) = . (3.3) X/P T 2

We also rely on a result of Zak [25, Corollary 7.4]:

N ∨ Theorem 3.4.3. If X is a smooth nonlinear subvariety of P , then dim(X ) ≥ dim(X).

N 3N−2 Suppose again that i : X,→ P is a smooth subvariety of dimension n ≥ 4 , and let m = N − n. By Theorem 3.4.3, dim(X∨) ≥ n, therefore 0 ≤ def(X) ≤ m − 1. Suppose further that def(X) > 0, then by Theorem 3.4.1 def(X) ≡ n mod 2. From (3.3), after taking into account the twist with [23, Example 3.2.2], we have

N − m − def(X) c = . 1 2

Now assume there exist integers Bc1,j depending only on c1 for j = 2, . . . , m such that each cj ≤ Bc1,j. We then obtain a brute-force algorithm which can be used to prove the duality defect N conjecture in the case of codimension m subvarieties of P . This is a generalized version of the algorithm given by Oaland [49, Kapittel 5] for the codimension 3 case of the conjecture.

3N−2 Algorithm 1. Input: integers m ∈ Z≥3, N ∈ Z≥10 such that N − m ≥ 4 Output: True or False

56 • for each r = 1, . . . , m − 1 with r ≡ N − m mod 2:

N−m−r • set c1 = 2 m−1 • for each tuple (c2, . . . , cm) ∈ Z≥0 such that each cj ≤ Bc1,j:

• if sj(c1, . . . , cm) > 0 for j = 0, . . . , n − r and sj(c1, . . . , cm) = 0 for j = n − r + 1, . . . , n: • return False

• return True

Here the algorithm returns True when there are no problematic Chern numbers cj satisfying N the requisite positivity conditions for a positive defect subvariety of codimension m in P , and thus the conjecture is true for that case. However, if problematic Chern numbers are found, we have not made any guarantee that they must come from a positive defect subvariety, and so the truth of the conjecture in that case falls outside the scope of this algorithm and the test is inconclusive. 2 To apply this algorithm, we need to find bounds Bc1,j. Oaland [49] provides the bound Bc1,2 = c1 by using the numerical nonnegativity of the Schur polynomials in the Chern classes of NX/PN (−1), 2 though this may also be obtained from the fact that s2 = c1 −c2. The idea to use Schur polynomials to obtain further bounds appears to be implicit in Oaland’s work [49, 4.3 Kodimensjon større enn

3]; however, no other bounds are provided and instead the possibilities for c3 given a choice of c1, c2 in the codimension 3 algorithm presented there are found as the nonnegative integer roots of sn(c1, c2, c3) ∈ Z[c3]. By continuing with this idea to use the nonnegativity of the Schur polynomials we are able to find bounds Bc1,j for every j ≥ 2.

N 3N−2 Lemma 3.4.4. Suppose i : X,→ P is a smooth subvariety of dimension n ≥ 4 with N −n ≥ 2. j j Then we have cj ≤ c1 for every j ≥ 2. That is, we may take Bc1,j = c1 for every j ≥ 2.

Proof. Let 2 ≤ j ≤ N − n. If λ = (j − 1, 1, 0,..., 0) represents the partition λ1 = j − 1 ≥ λ2 = 1 ≥

λ3 = 0 ≥ ... ≥ λj = 0 of j, then by Theorem 3.2.2 and Lemma 3.3.7 we have

∆λ(NX/PN (−1)) = c1(NX/PN (−1))cj−1(NX/PN (−1)) − cj(NX/PN (−1)) j ≥ = (c1cj−1 − cj)h ∈ An−j(X).

j Therefore deg(∆λ(NX/PN (−1))) ≥ 0 and so c1cj−1 − cj ≥ 0. Thus by induction on j, cj ≤ c1 for every j = 2,...,N − n.

57 Interestingly, these inequalities also follow from a recent result of Huh [38, Theorem 21] which imposes restrictions on the coefficients of the class of a subvariety of a product of projective spaces. One case of Huh’s result is the following:

N N Theorem 3.4.5. If Y is a subvariety of P × P and we write

X dim(Y )−j j N N [Y ] = aj[P × P ] ∈ A(P × P ) j for some aj ∈ Z, then the aj are nonnegative and form a log-concave sequence with no internal 2 zeros. Here log-concavity refers to the property that aj ≥ aj−1aj+1 for each j.

N+1 If F denotes the kernel of the surjection OX → NX/PN (−1), the Chern classes of NX/PN (−1) are the Segre classes of F , and since F ∨ is globally generated, together with Lemma 3.3.2 this shows that the

cj deg(X) = deg(cj(NX/PN (−1))) = deg(sj(F )) form a log-concave sequence with no internal zeros, and thus so too do the cj. In particular, 2 j cj ≥ cj−1cj+1 for every j = 1,...,N − n − 1. The inequalities cj ≤ c1 then follow by an induction argument.

N Remark 3.4.6. If X is a smooth subvariety of P for arbitrary N, then by Holme [35, Theorem 0 0 4.2] we have that δj(X) = δj+1(X ) for each j, where X is any general hyperplane section of X, N−1 0 treated as a subvariety of P . So if X has duality defect greater than 1, X will also be a positive defect subvariety. By Theorem 3.4.1, if X has positive duality defect, and if dim(X) is even, then so too is def(X), and thus def(X) ≥ 2. This means that to apply Algorithm 1 to a particular codimension, it suffices to check only half the possible N. For example, in the codimension 3 case one only needs to apply the algorithm successfully to an even N in order to prove the conjecture for N + 1 also.

Oaland [49, 6.2 Udata for kodimensjon 3] has run Algorithm 1 for codimension 3 subvarieties in N P for N = 10, 12,..., 140, which by Remark 3.4.6 proves the conjecture for N = 10, 11,..., 141. We have reproduced Oaland’s results by running Algorithm 1 using the SageMath computer algebra system [52], and have computed slightly further from N = 142, 144,..., 200. We have also run the algorithm in the codimension 4 and 5 cases for N = 14, 15, 17, 19,..., 47, 49 and N = 18, 20, 22

58 respectively. The cases N = 13, and N = 16, 17 are still of interest to the duality defect conjecture in these codimensions, but our current methods restrict us to N satisfying the constraint of Lemma 3.3.7. Altogether we have the following.

N Theorem 3.4.7. The duality defect conjecture is true for P in:

1. the codimension 3 case when N = 10, 11,..., 201,

2. the codimension 4 case when N = 14, 15,..., 50,

3. and in the codimension 5 case when N = 18, 19,..., 23.

Using the inequalities for the cj, we are able to obtain bounds for the degrees of positive defect subvarieties satisfying the constraints of Lemma 3.3.7 in terms of the duality defect and ambient space dimension:

N Theorem 3.4.8. If X is a smooth nonlinear codimension m subvariety of P with duality defect 3N−2 r > 0 and N − m ≥ 4 , then

m j X N − m − r  deg(X) ≤ . 2 j=0

Proof. Suppose we are given such an X, and let the cj be defined as they were at the beginning of m this section. By the self-intersection formula, cm(NX/PN ) = deg(X)h . From another application of ∼ m [23, Example 3.2.2], as NX/PN = NX/PN (−1)⊗OX (1), we obtain cm(NX/PN ) = (1+c1 +...+cm)h . N−m−r Therefore deg(X) = 1 + c1 + ... + cm. Since X has duality defect r > 0, c1 = 2 , and so by applying the bounds Bc1,j we obtain the desired inequality.

Results giving sufficient conditions for a smooth projective variety to be a complete intersection in terms of bounds on the degree of the variety have been established previously, such as that of Bertram et al. [11, Corollary 3] and Holme and Schneider [37, Theorem 5.1]. In particular, the result N N of Bertram et al. shows that any smooth subvariety X of P of codimension m with deg(X) ≤ 2m is a complete intersection. A stronger result of this form, in combination with Theorem 3.4.8, could potentially give a proof of the duality defect conjecture within the constraints of Lemma 3.3.7.

59 3.5 Duality defect in codimension 3

Using an observation about the positivity of a particular family of homogeneous order three integer linear recurrence sequences, we prove the duality defect conjecture for the codimension 3 N case when the dimension of the projective ambient space P is odd. This restriction to the case of odd ambient space dimension ensures that the possibilities for counterexamples that we must rule out are subvarieties with duality defect 2, a fact which makes the corresponding number-theoretic question more tractable.

N Theorem 3.5.1. The duality defect conjecture in the codimension 3 case is true for P when N is odd.

N Let i : X,→ P be a smooth nonlinear positive defect subvariety of codimension 3, where N ≥ 10 is odd. By Theorems 3.4.1, 3.4.3, we see that def(X) = 2. Let n = N − 3. By Larsen’s theorem, see Lemma 3.3.7, we can assume

2 3 c(NX/PN (−1)) = 1 + c1h + c2h + c3h and ∨ 2 n s(NX/PN (−1) ) = 1 + s1h + s2h + ... + snh

∗ N where h = i ([H]) for a hyperplane H ⊆ P , for some sj, cq ∈ Z≥0. Here the sj form part of the 2 recurrence sequence s0 = 1, s1 = c1, s2 = c1 − c2,

sj = c1sj−1 − c2sj−2 + c3sj−3 for j > 2. Furthermore, δj(X) = deg(X)sn−j for each j.

Since def(X) = 2, δ0(X) = δ1(X) = 0, and δj(X) > 0 for j > 1. Thus we see sn, sn−1 = 0, and sj > 0 for j = 0, . . . , n − 2. From these properties it follows that c1, c2, c3 > 0 as well. To prove our theorem, we investigate the positivity of this sequence. Note we get the same later terms if we start with initial conditions u0 = 0, u1 = 0, u2 = 1 instead. That is, it suffices to study the sequence

uj = c1uj−1 − c2uj−2 + c3uj−3 with u0 = 0, u1 = 0, u2 = 1, since we have uj+2 = sj for each j. The characteristic polynomial of 3 2 this sequence is ρ = t − c1t + c2t − c3.

60 From the results of Section 3.4, we know the codimension 3 duality defect conjecture is true for N = 10, 11,..., 201. Therefore, if we could conclude via a purely number-theoretic result that the only possibilities for a recurrence sequence with the properties of (uj)j∈Z≥0 are those with n ≤ 198, we would have proved our theorem. In fact, a much more precise result of this type is true.

Lemma 3.5.2. Let (uj)j∈Z≥0 be the homogeneous integer linear recurrence sequence defined by the initial conditions u0 = 0, u1 = 0, u2 = 1, and uj = c1uj−1 − c2uj−2 + c3uj−3 for j > 2, where c1, c2, c3 ∈ Z>0. Suppose there exists an m ∈ Z>2 with um = um+1 = 0 and uj > 0 for j = 2, . . . , m − 1. Then m = 4 or 6.

Proof. First note that m = 3 is not possible, as u3 = c1 which is assumed to be positive. Let   c1 −c2 c3 A =  1 0 0  . 0 1 0

Then for every j > 2, we have     uj−1 uj A uj−2 = uj−1 . uj−3 uj−2 So in particular, 1 1 m A 0 = d 0 , 0 0 where d = c3um−1 ∈ Z>0. This reflects the fact that the existence of m forces the sequence (uj)j∈Z≥0 to repeat up to multiplication by d every m steps. Note       1 u3 u4 0 ,  1  , u3 0 0 1 are linearly independent vectors, and as they are all eigenvectors associated to the eigenvalue d of Am, if we let 1 B = 1 A, d m m then B = id3, the identity element of the multiplicative group of 3 × 3 invertible real matrices, GL(3, R). By the minimality of m, B has order m as an element of GL(3, R). The eigenvalues of m m m m A are exactly the α1 , α2 , α3 , where

3 2 (t − α1)(t − α2)(t − α3) = det(t id3 −A) = t − c1t + c2t − c3 = ρ

61 is the factorization over C of the characteristic polynomial of A, which is also the characteristic polynomial of the sequence (uj)j∈Z≥0 . Here the αj must be distinct otherwise the positivity of m either the uj or the cj is violated. Therefore, as d is the only eigenvalue of A , ρ must divide tm − d. 1 3 3 3 If d m 6∈ Z, then ρ and t − d m are irreducible elements of Z[t]. Here d m = c3 ∈ Z>0. Then 1 3 3 since d m is a root of ρ, we see ρ must be equal to t − d m , which is impossible since we assume c1, c2 > 0. 1 This implies d m ∈ Z, and therefore B is an element of order m in GL(3, Q). Such matrices are well understood via elementary methods, and in particular the possible finite orders of elements of

GL(3, Q) are 1, 2, 3, 4, 6 [41]. Thus the only possibilities for m are 4, 6.

This result is sharp in that both possibilities for m can occur, for instance:

Example 3.5.3.

m = 4: take c1 = 3, c2 = 9, and c3 = 27. Then

(uj)j∈Z≥0 = (0, 0, 1, 3, 0, 0, 81, 243, 0, 0, 3561, 19683, 0, 0,...),

m = 6: take c1 = 4, c2 = 8, and c3 = 8. Then

(uj)j∈Z≥0 = (0, 0, 1, 4, 8, 8, 0, 0, 64, 256, 512, 512, 0, 0,...).

In view of Theorem 3.4.1, we have restricted ourselves to the case when the ambient projective space has odd dimension to simplify the corresponding number-theoretic problem. As demonstrated by Lemma 3.5.2 it is a relatively simple matter to understand the possibilities for homogeneous order three integer linear recurrence sequences that have two consecutive zeros in addition to their initial conditions. However, by Remark 3.4.6, the entire codimension 3 duality defect conjecture would follow if we could prove it for the case of even ambient space dimensions instead. The number-theoretic question in this case appears to be far more complicated, closer to the general problem of finding the zeros of linear recurrence sequences, a problem which has received much study but remains difficult [22, Chapter 2]. Nevertheless, there are results in this direction, such as one due to Mignotte et al. [46, Theorem 4] which gives a computable bound for the indices of zeros of order three recurrence sequences subject to a certain nondegeneracy condition. It would be interesting to know if such a result in

62 conjunction with the positivity constraints presented here for recurrence sequences representing positive defect subvarieties is enough to prove more cases of the duality defect conjecture.

63 CHAPTER 4

A RELATIVE SEGRE ZETA FUNCTION

4.1 Introduction 4.1.1 Motivation

For a homogeneous ideal I ⊆ k[x0, . . . , xn] where k is any field, one can consider the Segre class n n s(Z, P ) of the closed subscheme i : Z := Proj(k[x0, . . . , xn]/I) ,→ P defined by I. This class lives in the Chow group A∗(Z) of Z and is a central object in Fulton-MacPherson intersection theory [23], n containing information about the embedding of Z into P . As it is usually the case that A∗(Z) n n ∼ n+1 lacks a simple presentation, one often considers the pushforward i∗s(Z, P ) ∈ A∗(P ) = Z[t]/(t ). Key information from the Segre class carries over to its pushforward and may be extracted from n the coefficients of the minimum degree integer polynomial representing i∗s(Z, P ). Consequently the problem of computing this polynomial has received a substantial amount of attention in the past several years [3] [21] [29] [34] resulting in a number of algorithms to compute the Segre class of such a Z, including one currently implemented in the CharacteristicClasses package of the Macaulay2 computer algebra system [27].

For N ≥ n, if we denote by IN the extension of the ideal I to the larger polynomial ring N k[x0, . . . , xN ] via the inclusion, we may consider the closed subscheme iN : ZN ,→ P cut out by N n N IN . We call this the cone over Z in P since P embeds into P as the linear subvariety cut out N n by xn+1, . . . , xN , and on closed points Z is the image of ZN by the linear projection of P onto P away from the subvariety defined by x0, . . . , xn. It is natural to ask whether the Segre classes of the members of this infinite sequence of cones are all related.

The answer was recently given by Aluffi [5] in the Segre zeta function ζI (t) of I, which is defined to be the power series X j ζI (t) = ajt ∈ Z[[t]] j≥0 satisfying N ζI (HN ) = (iN )∗s(ZN , P )

64 N for every N ≥ n, where HN ∈ A∗(P ) is the hyperplane class. This is well-defined and the series turns out to be rational [5, Theorem 5.8]. Rationality implies that in order to know the pushforward of the Segre class of each of the cones in this infinite N sequence it suffices to know just one of the (iN )∗s(ZN , P ) for a large enough N. When one picks a homogeneous generating set for I the rational expression equal to ζI (t) and the N required above can be described more explicitly. These facts, when used in conjunction with existing Segre n class algorithms, can be used to significantly improve the speed of computing i∗s(Z, P ) in some cases [5, Example 5.11]. The Segre zeta function itself is a direct source of other invariants of projective schemes as well; for example, if Z is a smooth variety, the polar degrees of Z appear in t the coefficients of ζI (− 1+t ) [5, Proposition 6.2]. The purpose of this chapter is to explore a generalization of the Segre zeta function. The N n linear projection mentioned earlier is the rational map P 99K P corresponding to the inclusion k[x0, . . . , xn] ,→ k[x0, . . . , xN ]. Alternatively, one may derive this rational morphism from the ⊕(N+1) ⊕(n+1) N n natural surjection of bundles O{pt} → O{pt} over a point where P , P are interpreted as the projective bundles associated to these vector bundles. Therefore, more generally, one could try to define cones in all projective bundles over a smooth variety that dominate a given one, where we say one projective bundle dominates another if there is a surjection of the underlying vector bundles. In this case that surjection of bundles induces a rational map of projective bundles that is fiberwise a linear projection. This is the nature of the generalization we establish here; we will construct a relative Segre zeta function that describes the Segre classes of the cones in these larger projective bundles. Note that in the absolute case, only projective spaces are considered, and these organize into an infinite sequence of projections from progressively larger projective spaces. Z

n n+1 n+2 n+3 n+4 n+5 P P P P P P ··· However, an essential feature of our more general context is that there can be many non- isomorphic projective bundles of the same dimension dominating any one projective bundle P (E) [20, Theorem 9.5], and these make up an infinite tree rooted at P (E), with possibly many branches. When we have a sufficiently nice closed subscheme Z of P (E) our relative Segre zeta function will describe the Segre class of the cones over Z in all these larger projective bundles.

65 More specifically suppose we have such a rational map P (F ) 99K P (E) of projective bundles over a smooth variety X, and suppose i : Z,→ P (E) is a closed subscheme of P (E) which is the zero scheme of a section s of a bundle G on P (E). Then provided the rank of G is less than that of E and provided there exists a bundle on P (F ) and a section of this bundle sufficiently “compatible” with G and s, a notion which we will make precise below, we may define a subscheme Zˆ of P (F ) which will be the cone over Z in a sense that generalizes the cones of the absolute case. The problem of finding compatible bundles and sections seems independently interesting and is related to the difficulties with extending bundles from open subschemes. At least when G splits as a sum of line bundles, such an extension may always be found.

In this situation we define the relative Segre zeta function to be a formal power series ζG,s(t) with coefficients from A∗(X), with the property that when we evaluate it at the tautological class c1(OP (E)(1) ∩ [P (E)] of A∗(P (E)) we obtain the pushforward of s(Z,P (E)) to A∗(P (E)), and we obtain the pushforward of s(Z,Pˆ (F )) when we evaluate ζG,s(t) at the tautological class of

A∗(P (F )). Furthermore this series is rational by construction,

P (t) ζ (t) = , G,s Q(t) for some polynomials P (t),Q(t) ∈ A∗(X)[t] depending only on the representations of the classes c(G) ∩ i∗s(Z,P (E)) and c(G) ∩ [P (E)] in A∗(P (E)). As an immediate consequence of our work we derive a Segre zeta function for products of projective spaces which has already been used in [6] as a device for expressing the Chern-Schwartz- MacPherson class of a smooth projective variety [6, §4]. In the following part of this section, we introduce needed notation and state our results precisely.

4.1.2 Statement of the results

Throughout we take our base field k to be algebraically closed and for us varieties are integral schemes separated and of finite type over k. Let E be a rank e algebraic vector bundle on a smooth, n-dimensional variety X. We denote by P (E) the projective bundle of lines in E. The analogy to choosing a homogeneous ideal in the projective space setting to specify a closed subscheme is choosing a bundle G and one of its sections and indeed every closed subscheme of a projective bundle over a smooth variety is the zero scheme of such a section [23, B.8.1]

66 If G is a bundle on P (E), s : P (E) → G any section of G, and s0 the zero section embedding of P (E) into G, then the zero scheme of s is the scheme Z completing the following fiber square:

Z := P (E) ×G P (E) P (E)

i s (4.1) s P (E) 0 G

Just like with the Chow rings of projective spaces, the Chow ring of P (E) has a nice description as a ring: ∼ e e−1 A∗(P (E)) = A∗(X)[t]/(t + c1(E)t + ... + ce(E)) where the isomorphism identifies c1(OP (E)(1))∩[P (E)] with t. For simplicity of notation, whenever we invoke this identification we will abbreviate the Chern class notation; in this isomorphism ci(E) refers to ci(E) ∩ [X]. By pushing forward the Segre class s(Z,P (E)) of Z in P (E) to P (E) via the embedding i and using the above isomorphism, we obtain a class we can describe as a polynomial in t with coefficients from A∗(X). Tautologically we may write

−1 i∗s(Z,P (E)) = c(G) ∩ (c(G) ∩ i∗s(Z,P (E))), which presents i∗s(Z,P (E)) as a sort of rational expression in t.

Definition 4.1.1. More precisely, let P (t) ∈ A∗(X)[t] be the lowest degree polynomial representing the class e e−1 c(G) ∩ i∗s(Z,P (E)) ∈ A∗(X)[t]/(t + c1(E)t + ... + ce(E)), and Q(t) the lowest degree polynomial representing the class c(G) ∩ [P (E)]. We define the Segre zeta function of G with respect to s to be the rational expression P (t) ζ (t) := ∈ A (X)[[t]]. G,s Q(t) ∗ Suppose we are given a surjection of bundles F → E. Such a surjection induces a rational map of the corresponding projective bundles φ : P (F ) 99K P (E) and we let U be the open subscheme of P (F ) where φ is a morphism. To define what we mean by a cone over Z in P (F ), we try to find a bundle on P (F ) and section of said bundle which will serve as a kind of extension of G and s to P (F ). Specifically, we require a bundle Gˆ on P (F ) and a sections ˆ of Gˆ satisfying the following two conditions:

67 ˆ ∼ ∗ i there exists a bundle isomorphism G U = (φ U ) G, ii there is a sections ˆ : P (F ) → Gˆ making

ˆ =∼ ∗ G U (φ U ) G

sˆ U (φ )∗s U U

commute.

If such a bundle and section exist we define the cone in P (F ) over Z, denoted Zˆ, to be the zero scheme ofs ˆ. This scheme can still be thought of as a cone in a geometric sense; indeed the rational map φ is fiberwise a projection from a linear subvariety, and on closed points,

ˆ −1 Z = (φ U ) (Z).

Note that even when Z is reduced it does not suffice in general to define Zˆ by taking the reduced −1 scheme structure on the set-theoretic closure of (φ U ) (Z): in the absolute case for instance, the 1 2 closed subscheme of P (x : y) defined by the ideal (x , xy) is reduced but the cone cut out by the 2 same equations in P (x : y : z) is not. Our main result is:

Theorem 4.1.2. Suppose g < e, let F → E be any surjection of bundles on X, and let φ : P (F ) 99K P (E) be the induced rational map. If Gˆ is a bundle on P (F ) with section sˆ which satisfy properties (i), (ii) above, then letting ˆi : Z,ˆ → P (F ) denote the zero scheme of sˆ, we have

ˆ ˆ i∗s(Z,P (F )) = ζG,s(c1(OP (F )(1)) ∩ [P (F )]).

In other words, when the criteria of Theorem 4.1.2 are satisfied, the “same” rational expression that gives us i∗s(Z,P (E)) also gives ˆi∗s(Z,Pˆ (F )). The rank constraint is necessary to ensure that the representations for c(G)∩i∗s(Z,P (E)) and c(G)∩[P (E)] are unaffected by the relation defining the presentation of A∗(P (E)).

We in fact know a little more about the numerator of ζG,s(t). The highest dimension term of c(G) ∩ i∗s(Z,P (E)) is the top dimension term of i∗[Z], where [Z] is the fundamental class of Z in

A∗(Z), see [23, §1.5, Example 4.3.4]. We will show in Section 4.2 that the lowest dimension term of c(G) ∩ i∗s(Z,P (E)) is cg(G) ∩ [P (E)]. We collect these facts into an auxiliary proposition.

68 j d Proposition 4.1.3. For a ∈ A (X), we call j +d the total degree of the monomial at ∈ A∗(X)[t]. We have:

(a) the term of P (t) of lowest total degree is the minimal degree polynomial in A∗(X)[t] repre-

senting the top dimension part of i∗[Z], which is of total degree codim(Z),

(b) the term of P (t) of highest total degree is the minimal degree polynomial representing cg(G) ∩ [P (E)], which is of total degree g, and this is the same as the term of highest total degree of the denominator Q(t).

By using facts about the extension of reflexive sheaves, we note that Gˆ is unique up to isomor- phism, and show that such a bundle and section always exist at least when G splits as a sum of line bundles. Additionally, every closed subscheme of P (E) can be expressed as the zero scheme of a section of a bundle that splits into a sum of line bundles.

Proposition 4.1.4. Consider a surjection of bundles on X, F → E and let G be any bundle on P (E) with section s. Suppose that the rank e of E is ≥ 2.

(a) Any bundle Gˆ with section sˆ satisfying conditions (i) and (ii) of Theorem 1.2 is unique up to isomorphism.

(b) If G splits as a sum of line bundles then such a Gˆ and sˆ exist.

(c) Every closed subscheme Z of P (E) is the zero scheme of a section of a direct sum of line bundles.

(d) If Z is the zero scheme of a section of a bundle as in (b), (c), and if the rank of that bundle is < e, then Theorem 4.1.2 may always be applied.

So for instance, if one is able to choose fewer than e effective Cartier divisors on P (E) whose scheme-theoretic intersection is Z, then this is a situation where we may always successfully apply Theorem 4.1.2, taking G to be the sum of line bundles corresponding to the chosen divisors. The case where e = 1 is trivial in the sense that the Segre zeta function becomes unnecessary.

When e = 1, P (E) = X, and any rational map P (F ) 99K P (E) commuting with the projections to X extends to the projection map q : P (F ) → X. So if we had a closed subscheme Z of P (E) = X, the subscheme q−1(Z) of P (F ) would be the canonical choice for the cone over Z in P (F ), and we can use q to directly relate our Segre classes: q∗s(Z,P (E)) = s(q−1(Z),P (F )) [23, Proposition 4.2 (b)].

69 Finally, we may reduce the ambient space dimension of the projective bundle we are working with if given the existence of a sufficiently transverse codimension one subbundle of E.

Proposition 4.1.5. Suppose l : E0 ,→ E is a subbundle of rank e0 = e − 1, and suppose g < e0. Then l induces an embedding l : P (E0) ,→ P (E), and l∗G and l∗s are a bundle and section pair on P (E0) which cut out a subscheme i0 : Z0 ,→ P (E0). If P (E0) does not contain any of the supports of the irreducible components of the normal cone of Z in P (E), then in this case

ζG,s(t) = ζl∗G,l∗s(t).

This chapter is organized as follows. Sections 4.2 and 4.3 are dedicated to the proofs of Theorem 4.1.2, Proposition 4.1.4, respectively, and Proposition 4.1.3 will be also be proven in Section 4.2. In Section 4.4 we discuss the special situation where the surjection of bundles we are working with splits, F = E ⊕L, for some line bundle L on X, in which case Theorem 4.1.2 has a more basic proof. There we also prove Proposition 4.1.5. In Section 4.5 we show how our generalization specializes to the Segre zeta function of the absolute case, and as an application of the relative Segre zeta function we derive a Segre zeta function for products of projective spaces.

4.2 Proof of the main result

As in Section 4.1.2, let φ : P (F ) 99K P (E) be a rational map of projective bundles on a smooth variety X, induced by a surjection of the underlying bundles. Let U be the open subscheme of P (F ) where φ is a morphism, and let e = rank(E), f = rank(F ). Following Aluffi [5], we say that a bundle Gˆ on P (F ) is compatible with a bundle G on P (E) if condition (i) of Section 4.1.2 is ˆ ∼ ∗ satisfied, that is, if there is an isomorphism G U = (φ U ) G. Likewise, if s is a section of G, then a sections ˆ of Gˆ is compatible to s if condition (ii) is satisfied. We organize the proof of Theorem 4.1.2 as a sequence of lemmas.

Lemma 4.2.1.?? The hyperplane bundle on P (F ), OP (F )(1), is compatible with that of P (E).

Proof. Note that if E, F are the locally free sheaves of sections of E, F , respectively, then we have E = Spec(Sym(E∨)), F = Spec(Sym(F ∨)), and P (E) = Proj(Sym(E∨)),P (F ) = Proj(Sym(F ∨)). We can find a cover of X with affine open subschemes small enough so that the restrictions of E, F to each are simultaneously free. Let W = Spec(A) be one such subscheme.

70 Then

P (E)W := P (E) W = Proj(A[x1, . . . , xe]),P (F )W := P (E) W = Proj(A[y1, . . . , yf ]).

The surjection FW → EW corresponds to a degree-preserving, injective homomorphism of graded rings, A[x1, . . . , xe] ,→ A[y1, . . . , yf ]. This in turn defines the rational map

φ : P (F )W P (E)W . U∩P (F )W 99K

In this situation, the restrictions of the sheaves of sections of the hyperplane bundles are iso- morphic over U, that is,

∗ ∼ (φ ) OP (E)(1) = OP (F )(1) U∩P (F )W P (E)W U∩P (F )W

[32, Proposition II.5.12 (c)]. These local isomorphisms are compatible with restriction and thus glue together to yield an isomorphism globally:

∼ ∗ OP (F )(1) U = (φ U ) OP (E)(1).

Let K be the kernel of F → E which is thus a subbundle of F . The indeterminancy locus of φ is then P (K) ,→ P (E), and we may resolve φ by blowing up P (F ) along P (K). This produces the commutative diagram:

BlP (K) P (F ) π λ

φ P (F ) P (E)

q p X

Here π is a proper birational map, λ is the induced map to P (E), and p, q are the projections from P (E),P (F ) to X, respectively. The map λ realizes the blow up as a projective bundle over P (E) (see [19, Proposition 9.3.2] for the absolute case) and so it is in particular a flat morphism.

This gives us a way to move a class from A∗(P (E)) to A∗(P (F )). Following Aluffi [5] once more, we define for a class α ∈ A∗(P (E)) the join of α with P (K) to be

∗ α ∨ P (K) := π∗λ α ∈ A∗(P (F )).

71 Lemma 4.2.2.?? Let d < e. Then

d d (c1(OP (E)(1)) ∩ [P (E)]) ∨ P (K) = c1(OP (F )(1)) ∩ [P (F )].

−1 Proof. Let V denote the complement of the exceptional divisor in BlP (K) P (F ), so V = π (U). By the previous lemma, the O(1) bundles on P (F ),P (E) are compatible, and thus we obtain an isomorphism of bundles on the blow-up BlP (K) P (F ),

∗ ∼ ∗ (λ OP (E)(1)) V = (π V ) (OP (F )(1) U ).

In particular, this means these two bundles have the same first Chern class. By the commutativity of the blow-up diagram and by the functorial properties of Chern classes, if j : V,→ BlP (K) P (F ) denotes the open immersion, we have

∗ ∗ d ∗ d j (c1(π OP (F )(1)) ∩ [BlP (K) P (F )] − λ (c1(OP (E)(1)) ∩ [P (E)])) = 0.

From Fulton [23, Proposition 1.8], the sequence

∗ l∗ j A∗(D) A∗(BlP (K) P (F )) A∗(V ) 0 is exact, where l : D,→ BlP (K) P (F ) denotes the exceptional divisor. So knowing that

∗ d ∗ d ∗ c1(π OP (F )(1)) ∩ [BlP (K) P (F )] − λ (c1(OP (E)(1)) ∩ [P (E)]) ∈ ker j tells us this class is equal to the pushforward by l of a class α ∈ An+f−1−d(D).

Thus as dim P (K) = n + f − e − 1, so long as d < e, we see that π∗l∗α = 0. Therefore, we may conclude by the projection formula for Chern classes [23, Theorem 3.2 (c)].

A completely analogous argument establishes:

Lemma 4.2.3. If G is a rank g < e bundle on P (E), Gˆ a compatible bundle to G on P (F ), then (c(G) ∩ [P (E)]) ∨ P (K) = c(Gˆ) ∩ [P (F )].

∼ e e−1 The isomorphism A∗(P (E)) = A∗(X)[t]/(t + c1(E)t + ... + ce(E)) is defined so that the d d ∗ expression αt corresponds to c1(OP (E)(1)) ∩ p α, for any α ∈ A∗(X). Likewise for A∗(P (F )). This, the previous lemma, the commutativity of the blow-up diagram, and the projection formula for Chern classes together gives us the following:

72 d d Lemma 4.2.4. If d < e, then for any αj ∈ A∗(X), (αdt + ... + α0) ∨ P (K) = αdt + ... + α0.

Next, assuming we have a rank g bundle G on P (E), a section s of G, and by setting i : Z,→ P (E) to be the zero scheme of s, we show:

Lemma 4.2.5. The class

e e−1 c(G) ∩ i∗s(Z,P (E)) ∈ A∗(X)[t]/(t + c1(E)t + ... + ce(E)) can be represented by a polynomial in A∗(X)[t] of degree ≤ g.

Proof. The key point to show here is that the class c(G) ∩ i∗s(Z,P (E)) has no homogeneous components of codimension > g. This is Fulton [23, Example 6.1.6], as the diagram (1) is the setup for intersecting the image of P (E) by s with the zero section embedding of P (E) in G; the zero section embedding is a regular embedding with normal bundle G. Let N = i∗G. To see why this is the case in more detail, note that by Fulton [23, B.5.7], there is an exact sequence ∗ 0 → OP (N⊕1)(−1) → r (N ⊕ 1) → ξ → 0 of bundles on P (N ⊕ 1), where r : P (N ⊕ 1) → Z is the projection map, and ξ is the universal rank g quotient bundle on P (N ⊕ 1).

The normal cone of Z in P (E), C := CZ P (E) [23, Chapter 4] embeds into N, and so we may view its projective completion P (C ⊕ 1) as a closed subscheme of P (N ⊕ 1). Then, by the sum formula for Chern classes we have

∗ r∗(c(ξ) ∩ [P (C ⊕ 1)]) = r∗(c(r (N ⊕ 1))s(OP (N⊕1)(−1)) ∩ [P (C ⊕ 1)]).

The expression ∗ c(r (N ⊕ 1))s(OP (N⊕1)(−1)) ∩ [P (C ⊕ 1)] is equal to ∗ X j c(r (N ⊕ 1)) ∩ ( c1(OP (N⊕1)(1)) ∩ [P (C ⊕ 1)]). j≥0 So by the projection formula,

X j r∗(c(ξ) ∩ [P (C ⊕ 1)]) = c(N ⊕ 1) ∩ r∗( c1(OP (N⊕1)(1)) ∩ [P (C ⊕ 1)]). j≥0

73 Finally, X j r∗( c1(OP (N⊕1)(1)) ∩ [P (C ⊕ 1)]) = s(Z,P (E)) j≥0 by definition of the Segre class, and c(N ⊕ 1) = c(N), so we have

r∗(c(ξ) ∩ [P (C ⊕ 1)]) = c(N) ∩ s(Z,P (E)).

Since X and thus P (E) are varieties, P (C ⊕ 1) is of pure dimension dim(P (E)) [23, B.6.6]. So as ξ has rank g, it is clear that c(N) ∩ s(Z,P (E)) has no terms of dimension < dim P (E) − g. Thus the class c(G) ∩ i∗s(Z,P (E)) ∈ A∗(P (E)) has no terms of codimension > g. Therefore, via the isomorphism ∼ e e−1 A∗(P (E)) = A∗(X)[t]/(t + c1(E)t + ... + ce(E)), the class c(G) ∩ i∗s(Z,P (E)) may be represented by a polynomial in t of degree ≤ g.

Note that the term of codimension g of c(G) ∩ i∗s(Z,P (E)) is actually i∗P (E) ·G P (E), the pushforward of the intersection product of P (E) with itself, where here one P (E) is the image by s of P (E) and the other the image of P (E) by the zero section embedding s0 : P (E) ,→ G [23, Proposition 6.1 (a)]. The intersection product respects rational equivalence, and so since the image of P (E) by s is rationally equivalent to the image of P (E) by s0, so we may assume that both P (E) are in fact the image by s0 of P (E), showing that the codimension g term of c(G) ∩ i∗s(Z,P (E)) must be the codimension g term of c(G) ∩ i∗s(P (E),P (E)), which is cg(G) ∩ [P (E)]. This takes care of part (b) of Proposition 4.1.3. Suppose now that we have G,ˆ s,ˆ and Zˆ also defined as in Theorem 4.1.2. The join operation allows us to express the class ˆi∗s(Z,Pˆ (F )) in terms of i∗s(Z,P (E)). In particular we have the following [5, Corollary 4.5].

Lemma 4.2.6. If g < e,

ˆi∗s(Z,Pˆ (F )) = s(Gˆ) ∩ ((c(G) ∩ i∗s(Z,P (E))) ∨ P (K)), where here s(Gˆ) = c(Gˆ)−1 is the Segre class of Gˆ.

So if we assume g < e, then the other lemmas show that the minimal degree polynomials in

A∗(X)[t] representing c(G) ∩ i∗s(Z,P (E)) and c(G) ∩ [P (E)] remain unchanged after taking the

74 join with P (K), and furthermore that (c(G) ∩ [P (E)]) ∨ P (K) = c(Gˆ) ∩ [P (F )]. By properties of the intersection product on A∗(P (F )) [23, Example 8.1.6],

(c(G) ∩ i∗s(Z,P (E))) ∨ P (K) s(Gˆ) ∩ ((c(G) ∩ i∗s(Z,P (E))) ∨ P (K)) = . c(Gˆ) ∩ [P (F )]

Therefore, the right-hand side of Lemma 4.2.6 is equal to ζG,s(c1(OP (F )(1)) ∩ [P (F )]), proving Theorem 4.1.2.

4.3 Extension of bundles and sections

Parts (a), (b) of Proposition 4.1.4 follow immediately from several results on the extension of reflexive coherent sheaves from open subschemes which we list here. Recall that a reflexive sheaf on a scheme Y is an OY -module which is isomorphic to its double dual via the canonical evaluation map. Locally free sheaves are reflexive in particular. We have the following.

Lemma 4.3.1. If Y is a smooth scheme, j : U,→ Y an open immersion with codim(Y \ U) ≥ 2, and G a reflexive coherent sheaf on U, then

• j∗G is coherent and reflexive, ∼ • (j∗G) U = G, ∼ ∼ • if F1, F2 are any reflexive coherent sheaves on Y with F1 U = F2 U , then F1 = F2.

These facts are established in greater generality by Hassett and Kov´acs[33, Proposition 3.6, Corollary 3.7]. Thus in our particular situation, the question of whether the bundle G on P (E) ∗ extends to a bundle on P (F ) is equivalent to whether the sheaf j∗(φ U ) G is locally free, where here G is the locally free sheaf of sections of G and j is the open immersion U = P (F ) \ P (K) ,→ P (F ). It seems to be an interesting question whether this is always true in our situation. In the case e−1 f−1 where X is a point, and so P (E) = P ,P (F ) = P are projective spaces, some related results m have been established. For instance, Kempf [40] has proven that a bundle G on P , m ≥ 2, is 1 m a sum of line bundles if and only if the cohomology groups H (P , H om(G, G)(−a)) vanish for m+1 m+1 all a > 0 and G extends to a bundle on P in the sense that there is a bundle on P which m m+1 restricts to G on P ⊆ P , though it turns out the cohomological requirement already implies the extendability of G, rendering it a redundant assumption [42]. It is also known that if G extends

75 M in this way to P for every M > m, then G has the same total Chern class as a direct sum of line bundles, and if the rank of G is 2, then in fact G must actually be a sum of line bundles [8]. 4 Furthermore on P there is the Horrocks-Mumford bundle, an indecomposable rank 2 bundle (its 2 4 5 total Chern class is 1 + 5H4 + 10H4 ∈ A∗(P )) that fails to extend to a bundle on P [39, Theorem 2.7]. In this work we avoid this uncertainty by focusing on bundles which split into direct sums of line bundles as these always extend in our sense. Every subscheme of P (E) may be realized as the zero scheme of a section of such a bundle. Indeed, as we mentioned in Section 4.1.2, every closed subscheme Z of P (E) is the zero scheme of a section of some bundle on P (E). By [32, Exercise III.6.8 (b)], this bundle injects into a sum of line bundles, and so we may construct a section of that split bundle whose zero scheme is Z. So if the goal is to associate a relative Segre zeta function to a closed subscheme Z of P (E), the only issue stymieing Theorem 4.1.2 is whether we can find a sum of line bundles defining Z of sufficiently small rank, specifically of rank < e. Direct sums of line bundles extend to P (F ) because of the following well-known fact for which we make note of a brief argument due to lack of appropriate reference.

Lemma 4.3.2. Let Y be a smooth variety, and U an open subscheme. Suppose L is a line bundle ˆ ˆ ∼ on U. Then there exists a line bundle L on Y with L U = L.

Proof. Due to the smoothness of Y there is a bijective correspondence between Weil and Cartier divisors on Y up to linear equivalence, and likewise for U. A line bundle L on U corresponds Pd to a Cartier divisor, which then in turn corresponds to some Weil divisor D := i=1 ai[Vi] in

Adim(U)−1(U). Here the Vi are codimension one subvarieties of U, and the ai ∈ Z. Form the Weil Pd ∗ divisor D := i=1 ai[Vi] in Adim(Y )−1(Y ). Then j D = D. Furthermore D corresponds to a Cartier divisor and thus a line bundle Lˆ on Y which restricts to L.

∗ Therefore in our original situation, for an invertible sheaf L on P (E), j∗(φ U ) L is also invertible ∗ by Lemma 4.3.1. The functor j∗(φ U ) is additive so if we have a direct sum of invertible sheaves on P (E) we may simply apply it to get a sum of invertible sheaves on P (F ).

A section s of a bundle G on P (E) is specified by giving a map OP (E) → G . Assuming ∗ ˆ ˆ ∗ j∗(φ U ) G =: G is locally free, to get a section of the corresponding bundle G, we apply j∗(φ U )

76 to OP (E) → G and note ∗ ∼ j∗(φ U ) OP (E) = OP (F ) by Lemma 4.3.1. This section will be compatible with s in the sense of property (ii) described in the introduction. Note that the condition that the rank of E is ≥ 2 in Proposition 4.1.4 is what allows us to satisfy the codimension constraint of Lemma 4.3.1 to get the desired uniqueness property. This finishes the proof of Proposition 4.1.4.

4.4 The case of subbundles 4.4.1 When F = E ⊕ L

A special situation worth describing is when the surjection of bundles φ : F → E in Theorem 4.1.2 splits by way of F being a sum of E with a line bundle L on X, F = E ⊕ L. That is, if

l : E,→ E ⊕ L = F

is the inclusion, φ ◦ l = idE. In this case Theorem 4.1.2 admits a lower-tech proof. ∗ That F = E ⊕ L implies there is a closed embedding l : P (E) ,→ P (F ) for which l OP (F )(1) =

OP (E)(1) [23, B.5.1]. Instead of the join operation defined in Section 4.2 we may use the Gysin pullback map ∗ l : A∗(P (F )) → A∗(P (E)) induced by this embedding to relate classes in A∗(P (F )),A∗(P (E)) [23, 6.2]. The embedding l realizes Z as an effective Cartier divisor in Zˆ and we will abuse notation using l∗ to also denote the Gysin map A∗(Zˆ) → A∗(Z). Note l(P (E)) is contained in the complement U of the projectivization of the kernel of the vector bundle map φ, P (0 ⊕ L). Thus the stipulation that the vector bundle surjection splits in the above sense means that

φ U ◦ l = idP (E) .

Here we abuse notation again, using φ to also refer to the induced rational map φ : P (F ) 99K P (E). Therefore if Gˆ is a bundle on P (F ) compatible with a rank g bundle G on P (E), we have l∗Gˆ =∼ G. Further, ifs ˆ is a section of Gˆ compatible with a section s of G, and if i : Z,→ P (E), ˆi : Z,ˆ → P (F ) are the zero schemes of s, sˆ, then we see l−1(Zˆ) = Z.

77 Since L is a line bundle, P (E) is an effective Cartier divisor in P (F ) via l. The intersection P (E) ∩ Zˆ = Z is transverse in the sense that P (E) does not contain the support of any irreducible component of the normal cone of Zˆ in P (F ) [23, B.5.3], and so we may apply the following property of Segre classes, see [4, Lemma 4.1]:

Lemma 4.4.1. Let Z,→ W be schemes, D a Cartier divisor on W such that D does not contain the support of any irreducible component of the normal cone of Z in W , CZ W . Then

s(Z ∩ D,D) = D · s(Z,W ).

With our Gysin map notation, Lemma 4.4.1 implies l∗s(Z,Pˆ (F )) = s(Z,P (E)). Gysin maps are compatible with proper pushfoward, see [23, Theorem 6.2 (a)], and so this applied to the fiber diagram

Z = Zˆ ∩ P (E) l Zˆ

i ˆi P (E) l P (F )

in conjunction with Lemma 4.4.1 yields

∗ l ˆi∗s(Z,Pˆ (F )) = i∗s(Z,P (E)).

The numerator of the relative Segre zeta function is

c(G) ∩ i∗s(Z,P (E)).

Notice that for any α ∈ A∗(P (F )),

l∗c(Gˆ) ∩ α = c(G) ∩ l∗α.

By Lemma 4.2.5, using the presentation for A∗(P (E)) again, this numerator may be represented by a polynomial of degree ≤ g in t. Furthermore, we have

∗ l (c(Gˆ) ∩ ˆi∗s(Z,Pˆ (F ))) = c(G) ∩ i∗s(Z,P (E)), and additionally ∗ l c1(OP (F )(1)) ∩ [P (F )] = c1(OP (E)(1)) ∩ [P (E)].

78 Thus if we assume that g < e as in Theorem 4.1.2, the same minimal degree polynomial in t that represents

c(Gˆ) ∩ ˆi∗s(Z,Pˆ (F )) also represents c(G) ∩ i∗s(Z,P (E)). Similarly c(Gˆ) ∩ [P (F )], c(G) ∩ [P (E)] both have no terms of codimension exceeding g, and so as

l∗c(Gˆ) ∩ [P (F )] = c(G) ∩ [P (E)], both classes may also be represented by the same minimal degree polynomial in t.

Thus the rational expressions ζG,s(t) and ζG,ˆ sˆ(t) are identical, and therefore

ˆ ˆ ζG,s(c(OP (F )(1)) ∩ [P (F )]) = i∗s(Z,P (F )), proving Theorem 4.1.2.

4.4.2 Reduction of the ambient space dimension

Suppose again we have a rank g bundle G on P (E), a section s of G with zero scheme i : Z,→ P (E), and suppose we have a rank e0 := e − 1 subbundle E0 ,→ E so that P (E0) meets Z transversally in the sense of Lemma 4.4.1. This time we are not assuming there is a surjection E → E0. Let l : P (E0) ,→ P (E) be the induced closed embedding and let i0 : Z0 ,→ P (E0) be the zero scheme of the section l∗s. Then as before, Lemma 4.4.1 shows

∗ 0 0 0 l i∗s(Z,P (E)) = i∗s(Z ,P (E )).

So with the assumption g < e0,

ζG,s(t) = ζl∗G,l∗s(t) completeing the proof of Proposition 4.1.5. The use of this result is as a means of simplifying the projective bundle one needs to work within in order to compute ζG,s(t). If one has a filtration of subbundles of E each of rank one less than the next but still exceeding g, Proposition 4.1.5 may be applied repeatedly to further reduce the projective bundle dimension. This is an analog of intersecting with hyperplanes in the absolute case, where the ability to reduce the ambient space dimension can have substantial practical benefits [5, Example 5.11].

79 4.5 Examples 4.5.1 Subschemes of projective space

The result of Theorem 4.1.2 generalizes the notion of the Segre zeta function of a homogeneous ideal introduced in Aluffi [5]. Using the notation of Section 4.1.2, we recover this special case when ⊕(n+1) n X = {pt}, and E = OX , so P (E) = P . n A homogeneous ideal of the homogeneous coordinate ring of P , say I ⊆ k[x0, . . . , xn], can be written as I = (F0,...,Fr) for some choice of homogeneous generating set {F0,...,Fr} of I. Each

Fj is a section of the bundle OPn (dj), where dj := deg(Fj). n So as the closed subscheme of P defined by I,

n i : Z := Proj(k[x0, . . . , xn]/I) ,→ P , is the scheme-theoretic intersection of the effective Cartier divisors (hypersurfaces) cut out by the

Fj, we see that Z is the zero scheme of a section s of the bundle r M G := OPn (dj). j=0 So in this context, the rank constraint of Theorem 4.1.2 becomes r + 1 < n. ∼ ∼ N+1 Furthermore, note that A∗(X) = Z and A∗(P (E)) = Z[t]/(t ). The total Chern class of G n Qr is then c(G) ∩ [P ] = j=0(1 + djHn) where

n n Hn = c1(OPn (1)) ∩ [P ] ∈ A∗(P ), the class of a hyperplane, and the Segre zeta function has the form P (t) ζG,s(t) = Qr , j=0(1 + djt) which coincides with the Segre zeta function ζI (t) defined by Aluffi [5] and recovers its rationality. More is known about the numerator of this rational function in general, as described in [5]. In n particular since we may obtain c(G) ∩ i∗s(Z, P ) as a pushforward of the total Chern class of the universal quotient bundle of P (G ⊕ 1), see the proof of Lemma 4.2.5, and since G is globally gener- n ated, it follows this class c(G)∩i∗s(Z, P ) is nonnegative [23, Example 12.1.7]. So the coefficients of n P (t) are nonnegative. Also, by Proposition 4.1.3, the highest codimension term of c(G)∩i∗s(Z, P ) is n r+1 cg(G) ∩ [P ] = d0 ··· drHn ,

80 r+1 so the leading term of P (t) is d0 ··· drt . ⊕(N+1) N ˆ For a larger projective space, given by F = OX , P (F ) = P , N ≥ n, we can take G = Lr ˆ ˆ N j=0 OPN (dj) which is compatible with G, and use the obvious compatible section. If i : Z,→ P is the zero scheme of this section, and if r + 1 < n then by Theorem 4.1.2,

ˆ ˆ N i∗s(Z, P ) = ζG,s(HN ) = ζI (HN ),

N where now HN ∈ A∗(P ) is the hyperplane class. P j If we express ζG,s(t) as a formal power series, j≥0 ajt ∈ Z[[t]], it is then clear that we truncate everything but the first N + 1 terms of this infinite series to obtain an integral polynomial in HN ˆ ˆ N equal to i∗s(Z, P ).

4.5.2 Subschemes of products of projective spaces

We may form a Segre zeta function for products of projective spaces and derive its properties as a convenient consequence of the relative Segre zeta function of Theorem 4.1.2. This formulation of the Segre zeta function has appeared in recent work where it is used to give a formula for the Chern-Schwartz-MacPherson class of a closed subscheme of a smooth projective variety [6]. The alternate proof of Theorem 4.1.2 given in Section 4.4 may be used to derive the relative Segre zeta function we will need here as well as the absolute Segre zeta function of the previous example. n m Consider the product P × P . Here we may realize this scheme simultaneously as both the projective bundle P (O⊕(m+1)) and P (O⊕(n+1)). Let p , q denote the projection maps to the first Pn Pm n m and second component of the product, respectively. n m Fixing coordinates, P (x0 : ... : xn), P (y0 : ... : ym), note that any closed subscheme i : n m Z,→ P × P may be written as the zero scheme of a finite collection of polynomials F0,...,Fr, bihomogeneous in the xj and the yj. For each j, let (aj, bj) be the bidegree of Fj. More precisely, each Fj corresponds to a section of the line bundle

∗ ∗ OPn×Pm (aj, bj) := pnOPn (aj) ⊗ qmOPm (bj).

So Z is the zero scheme of a section s of the bundle

r M G := OPn×Pm (aj, bj). j=0

81 n m The pushforward of its Segre class i∗s(Z, P × P ) is a class in the Chow ring

n m n+1 m+1 A∗(P × P ) = Z[s, t]/(s , t ),

n m where this isomorphism identifies s, t with the pullbacks of the hyperplane classes in P , P by pn, qm, respectively. n m There are unique polynomials P (s, t),Q(s, t) in s, t representing the classes c(G)∩i∗s(Z, P ×P ) n m and c(G) ∩ [P × P ] respectively, of degree < n + 1 in s and degree < m + 1 in t. In particular, Qr by the sum formula for Chern classes, Q(s, t) = j=0(1 + ajs + bjt), provided r < n, m.

Definition 4.5.1. We define the Segre zeta function of G with respect to s in this context to be

P (s, t) P (s, t) ζG,s(s, t) := = Qr ∈ Z[[s, t]]. Q(s, t) j=0(1 + ajs + bjt)

The bundle G is globally generated when the aj, bj ≥ 0, and so by analogous reasoning as in the previous example, the coefficients of P (s, t) are nonnegative. By Proposition 4.1.3, the highest n m degree term of the numerator with respect to total degree comes from cg(G) ∩ [P × P ] and so is

(a0s + b0t) ··· (ars + brt). 0 n m n m Denote by Hn,Hm ∈ A∗(P ×P ) the pullbacks of the hyperplane classes from P , P , by pn, qm respectively. Then we recover the Segre zeta function defined in Section 4.1.2 for both the case where we consider n × m to be the projective bundle P (O⊕m+1) and P (O⊕n+1) as ζ (H , t) P P Pn Pm G,s n 0 and ζG,s(s, Hm), respectively. Suppose N,M are integers with N ≥ n, M ≥ m. Then we have a closed subscheme ˆi : Z,ˆ → N M P (x0 : ... : xN ) × P (y0 : ... : yM ) which is the zero scheme of the sections ˆ of the bundle ˆ Lr N M G = j=0 OPN ×PM (aj, bj) on P × P corresponding to the Fj. If r < n, m, then all of our considerations establish that

ˆ ˆ N M 0 i∗s(Z, P × P ) = ζG,s(HN ,HM ).

For convenience, we collect these observations into a proposition.

Proposition 4.5.2. Let n, m, r ≥ 0, be integers with r < n, m and let F0,...,Fr be bihomogeneous polynomials in x0, . . . , xn, y0, . . . , ym of bidegrees (a0, b0),..., (ar, br). Take

r M G := OPn×Pm (aj, bj), j=0

82 and let s denote the section of G determined by the Fj. Then there is a formal power series of the form P (s, t) ζG,s(s, t) = Qr ∈ Z[[s, t]], j=0(1 + ajs + bjt) such that:

N M 1. For any N ≥ n, M ≥ m, if iN,M : ZN,M ,→ P × P is the zero scheme of the section of Lr j=0 OPN ×PM (aj, bj) determined by the Fj, then

N M 0 N M (iN,M )∗s(ZN,M , P × P ) = ζG,s(HN ,HM ) ∈ A∗(P × P ),

0 N M where HN ,HM the pullbacks of the hyperplane classes from P , P , respectively.

2. The coefficients of the polynomial P (s, t) ∈ Z[s, t] are nonnegative.

3. P (s, t) is of degree ≤ n in s, of degree ≤ m in t, and its term of highest total degree is

(a0s+b0t) ··· (ars+brt). Its lowest total degree term is of total degree codim(Z), and represents n m the top dimension part of the pushforward of the fundamental class [Z] in A∗(P × P ).

83 CHAPTER 5

EQUIVARIANT REALIZATION OF THE SEGRE ZETA FUNCTION

5.1 Introduction

∗ Let X be a pure-dimensional algebraic C-scheme. If X is equipped with a C -action, then for ∗ ∗ any C-scheme U with a free C -action such that the principal bundle quotient U → U/C exists as ∗ ∗ a scheme, a principal bundle quotient X × U → (X × U)/C also exists as a scheme, since C is ∗ C a special group [14, Proposition 23]. One may define the equivariant Chow group A∗ (X) of X by ∗ C setting Aj (X) = Aj+N−1(XC∗ ). Here we form XC∗ by choosing any N dimensional representation ∗ ∗ V for C with N > dim(X) − j, which has a dense open subset U on which C acts freely, and then ∗ N defining XC∗ = (X × U)/C . For instance, one could take U = C \{0} for sufficiently large N. This construction is independent of the representations used, so long as they satisfy the specified dimension constraint [14, Section 2.2]. ∗ From now on, suppose that X has been given the trivial C -action.

Lemma 5.1.1. We have

∗ C ∼ A∗ (X) = A∗(X)[~].

∗ C N Proof. Note Aj (X) = AN+j(X × P ), for sufficiently large N, as a special case of the definitions N ∼ of the equivariant Chow groups. By the structure theorem for projective bundles, AN+j(X ×P ) = ∗ LN N−1 k C k=0[X × P ] Ak+j(X). We can define a map A∗(X)[~] → A∗ (X) on monomials of the form ∗ k k N−1 k N C α~ where α ∈ Ar(X) by defining α~ 7→ α[X × P ] ∈ AN+(r−k)(X × P ) = Ar−k(X). This map extends linearly to a group isomorphism.

Pr j j For an element α = j=0 α ∈ A∗(X), where α denotes the homogeneous piece of codimension j, we define, for a choice of k ≥ r, the homogenization of degree k of α to be

r ∗ χ X k−j j C α = ~ α ∈ A∗ (X). j=0

84 ∗ This definition is compatible with morphisms of schemes equipped with trivial C -actions. That ∗ is, if Y is another algebraic scheme equipped with the trivial C action and f : X → Y a proper map, then we have the following:

Lemma 5.1.2. Denoting by f∗ both the equivariant and non-equivariant pushforward maps, we have

∗ k k C f∗(α~ ) = f∗(α)~ ∈ A∗ (Y ) = A∗(Y )[~], for any α ∈ A∗(X).

∗ ∗ C C Proof. By definition, the equivariant f∗ is defined so that f∗ : Aj (X) → Aj (Y ) is (f × idPN )∗ : N N AN+j(X × P ) → AN+j(Y × P ) for sufficiently large N. By first assuming α is homogeneous, we obtain the desired result immediately from the isomorphism defined in the proof of the previous lemma.

If π : E → X is a vector bundle of rank e + 1, and if E is equipped with the action of fiberwise ∗ C -dilation, with character the identity, then the mixing space of E with respect to this action,

EC∗ , is ∗ ∗ EC = p E ⊗ OPN (−1),

N where p : X × P → X is the projection onto the first component, for sufficiently large N. By definition of equivariant Chern classes [14, Section 2.4] we obtain:

∗ C χ Lemma 5.1.3. For any subvariety V of X, ce+1(E) ∩ [V ]C∗ = (c(E) ∩ [V ]) , where the homoge- nization is taken to degree e + 1 + codimX (V ), and [V ]C∗ is the equivariant fundamental class of V .

∗ ∗ C C N Proof. By definition, ce+1(E) ∩ [V ]C∗ = ce+1(EC∗ ) ∩ [V × P ]. By Fulton [23, Remark 3.2.3], ∗ e+1 j ∗ N ∗ C ∗ P ce+1(EC ) = j=0 c1(OPN (−1)) ce+1−j(p E). Since [V × P ] = p [V ], and c1(OPN (−1)) = ~, we ∗ C N Pe+1 j ∗ have by the functoriality of Chern classes that ce+1(EC∗ )∩[V ×P ] = j=0 ~ p (ce+1−j(E)∩[V ]) = (c(E) ∩ [V ])χ, as desired.

We may deduce by applying Fulton [23, Theorem 3.3] to the bundle π : EC∗ → XC∗ that ∗ C ∼ ∗ A∗ (E) = A∗(X)[~] via the pullback map π . If s : X → E is the zero section embedding, it is ∗ ∗ ∗ −1 C -equivariant, and we define s = (π ) , both in the equivariant and non-equivariant cases.

85 ∗ Now define a C -action on the bundle E ⊕ 1 by using the same fiberwise action as before on ∗ the first component and the trivial action on the second component. This induces a C -action on the projective bundle P (E ⊕ 1). Note that E embeds into P (E ⊕ 1) as the open subset that is the ∗ ∗ complement of P (E ⊕ 0). This open immersion is C -equivariant with respect to the C -actions on both schemes. ∗ Pr For a C -invariant cycle C = j=0 ajVj, aj ∈ Z, Vj subvarieties, of codimension k on E, we Pr can define its closure in P (E ⊕ 1) as C = j=0 aj[Vj], using the immersion E,→ P (E ⊕ 1). Then from Aluffi et al. [7, Proposition 2.7]:

∗ ∗ χ C Lemma 5.1.4. We have s [C]C∗ = (ShadowE⊕1([C])) ∈ A∗ (X) where the homogenization is taken to degree k.

Here, by the structure theorem for projective bundles, every class α of codimension k in Pf j ∗ k−j A∗(P (F )) for a vector bundle F → X of rank f + 1 may be written as j=0 c1(O(1)) q (α ) for k−j a unique choice of α of codimension k − j in A∗(X) where q : P (F ) → X is the projection. We define f X k−j ShadowF (α) = α ∈ A∗(X). j=0 By Aluffi et al. [7, Lemma 2.1 (i)], this definition satisfies:

Lemma 5.1.5. X j ShadowF (α) = c(F ) ∩ q∗( c1(O(1)) ∩ α). j≥0

5.2 Segre classes

Let N denote the dimension of X and let i : Z,→ X be a closed pure-dimensional subscheme the support of which does not contain any of the irreducible components of X. Suppose π : E → Z is any rank e + 1 vector bundle with the property that C := CZ X embeds into E, l : C,→ E, such that the induced embedding l : P (C) ,→ P (E) satisfies l∗O(1) = O(1). Endow Z with the ∗ trivial C -action, and E the fiberwise dilation action with character the identity. Then, using the notation of the previous section:

∗ χ Proposition 5.2.1. We have s [C]C∗ = (c(E) ∩ s(Z,X)) , where the homogenization is taken to degree codimE(C).

86 ∗ χ Proof. By Lemma 5.1.4, we have s [C]C∗ = (ShadowE⊕1([C])) , where the homogenization is taken to degree e + 1 − codimX (Z). By Lemma 5.1.5 ShadowE⊕1([C]) equals

X j X j c(E ⊕ 1) ∩ q∗( c1(O(1)) ∩ [C]) = c(E) ∩ q∗( c1(O(1)) ∩ [C]) j≥0 j≥0 where q : P (E ⊕ 1) → Z is the projection. Note that if p : P (C⊕1) → Z is the projection, then there is a natural embedding l : P (C⊕1) ,→ P (E⊕1) induced by the embedding P (C) ,→ P (E), such that p = q◦l. Note also that [C] = P (C⊕1) in P (E ⊕ 1) by Fulton [23, B.5.3]. So by the definition of the Segre class, and by the functoriality properties of Chern classes, we have

X j X j c(E) ∩ s(Z,X) = c(E) ∩ p∗ c1(O(1)) ∩ [P (C ⊕ 1)] = c(E) ∩ q∗ c1(O(1)) ∩ [P (C ⊕ 1)], j≥0 j≥0 which gives the desired result.

Due to our hypotheses on Z, we have

X j X j q∗ c1(O(1)) ∩ [P (C)] = p∗ c1(O(1)) ∩ [P (C)] = s(Z,X), j≥0 j≥0 where p : P (C) → Z, q : P (E) → Z are the projections. By Fulton [23, B.5.7], there is an exact sequence 0 → O(−1) → q∗E → ξ → 0 on P (E), where ξ is the universal rank e quotient bundle on P (E). This allows us to write

∗ q∗(c(ξ) ∩ [P (C)]) = q∗(c(q E)s(O(−1)) ∩ [P (C)]), by the Whitney sum formula. This is then equal to

X j c(E) ∩ q∗ c1(O(1)) ∩ [P (C)] = c(E) ∩ s(Z,X) j≥0 by the projection formula for Chern classes and our above considerations. By an analogous argument,

q∗(c(ξ) ∩ [P (C ⊕ 1)]) = c(E) ∩ s(Z,X),

87 where ξ is now taken to be the universal quotient bundle on P (E ⊕ 1). We have from Fulton [23, Proposition 6.1 (a)] that ∗ q∗(ce+1(ξ) ∩ [P (C ⊕ 1)]) = s [C].

∗ Since C is pure dimensional, this also shows q∗(ce(ξ) ∩ [P (C)]) = s [C]. ∗ In the equivariant setting, we may endow P (E) with the trivial C -action, and ξ with the fiberwise dilation action. Then from Lemma 5.1.3,

∗ ∗ C χ C ce (ξ) ∩ [P (C)]C∗ = (c(ξ) ∩ [P (C)]) ∈ A∗ (P (E)), with homogenization taken of degree e + codimP (E)(P (C)).

If we take the equivariant pushforward q∗ of this, we obtain by Lemma 5.1.2

∗ χ C (q∗(c(ξ) ∩ [P (C)])) ∈ A∗ (Z) where this homogenization is taken to degree codimP (E)(P (C)) = codimE(C), as the fibers of q are of dimension e. Thus combining all of our results, we conclude:

Proposition 5.2.2. In the ordinary setting,

∗ q∗(ce(ξ) ∩ [P (C)]) = s [C], and in the equivariant setting,

∗ C ∗ χ q∗(ce (ξ) ∩ [P (C)]C∗ ) = s [C]C∗ = (c(E) ∩ s(Z,X)) .

This justifies in a sense the fact that in the equivariant context our zero section pullback of the equivariant fundamental class of the normal cone is (c(E) ∩ s(Z,X))χ, whereas in the ordinary setting the pullback of the class of the normal cone is just one homogenous term of the class c(E) ∩ s(Z,X). Specifically, it expresses this fact as a consequence of the fact top equivariant Chern classes of bundles endowed with the fiberwise dilation action over bases with the trivial ∗ C -action represent total Chern class information.

Example 5.2.3. Let I be a homogeneous ideal of C[x0, . . . , xN ], with homogeneous generators N F1,...,Fr of degrees d1, . . . , dr, respectively. Take Z be the closed subscheme of P cut out by I, and for each j, let Xj be the closed subscheme cut out by Fj. Then we have the following fiber square:

88 N Z P δ ∆ Qr Qr N j=1 Xj j=1 P

N N ∗ r N The normal cone C = C of Z in is a subcone of the bundle δ (NQr Q ). If Z P P j=1 Xj j=1 P χ ∗ we take this to be our E, then our above results state that (c(E) ∩ s(Z,X)) = s [C]C∗ . Here c(E) ∩ s(Z,X) has an interpretation as the numerator of the Segre zeta function, see Aluffi [7].

89 CHAPTER 6

CONCLUSION

Chapter 2 was an exploration of the secant indices of projective subvarieties. We derived several properties that these sequences satisfy, and defined accessory sequences that appear more tractable to compute for some varieties and are conjecturally equal to the secant indices. Veronese and Segre varieties have nice enough embeddings into projective space so that computing the term- wise lower bounds for the secant indices of those varieties is possible, and we stated combinatorial problems equivalent to determining those bounds along with some explicit computations in the lower-dimension cases. For future work, it would be interesting to see if one could verify these (2) 2 2 2 conjectures for Veronese and Segre varieties, at least for cases beyond v3 (P ) and σ2,2(P × P ). Furthermore, the accessory sequences we have defined exist for all smooth projective subvarieties and are heavily dependent on the existence of sufficiently reducible hyperplane sections of the varieties. It would be interesting to find more varieties besides just the images of the Veronese and Segre embeddings admitting enough such sections to extract tractable combinatorial problems for computing the lower bounds for the secant indices. Our work in Chapter 3 proved infinitely many cases of the duality defect conjecture in the codi- mension 3 case in addition to deriving a degree bound that any counterexamples to the conjecture would have to satisfy. This pushes a method of Holme further by capitalizing on a little more number theory, specifically on the theory of the occurrence of zeros in linear recurrence sequences, see Lemma 3.5.2. One future goal would be to see how far this number theoretic approach may be pushed. The problem of determining the positivity of an arbitrary homogeneous linear recurrence sequence is one that is mostly open in the field of number theory; however, the recurrence sequences arising in the computation of the duality defect satisfy a number of positivity constraints. It seems reasonable to expect that these in conjunction with deeper, purely-number theoretic machinery may suffice to prove more cases of the duality defect conjecture. Chapters 4 and 5 concerned generalizations of Aluffi’s Segre zeta function. One question leftover from the work in Chapter 4 has to do with the extensibility of bundles in the sense required to

90 define cones over subschemes of projective bundles, see Proposition 4.1.4. For our purposes, the condition that a subscheme was the zero scheme of a section of a bundle of small enough rank that fully splits as a sum of lines bundles sufficed to ensure that a cone could be constructed. A problem which seems interesting is to determine for how many bundles can one find a suitably compatible bundle with which to define the cone over a given subscheme.

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95 BIOGRAPHICAL SKETCH

Grayson Jorgenson received a Bachelor of Science degree from Florida Institute of Technology in applied mathematics. After completing his PhD at FSU, he aims to pursue a career in artificial intelligence.

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