Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry Corey S

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2017 Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry Corey S. (Corey Scott) Harris

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COLLEGE OF ARTS & SCIENCES

EFFECTIVE METHODS IN INTERSECTION THEORY AND COMBINATORIAL

ALGEBRAIC GEOMETRY

COREY S. HARRIS

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

2017

Corey S. Harris defended this dissertation on April 7, 2017. The members of the supervisory committee were:

Paolo Aluﬃ Professor Directing Dissertation

Eric Chicken University Representative

Ettore Aldrovandi Committee Member

Kyounghee Kim 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

First and foremost I would like to thank my advisor, Professor Paolo Aluffi. Even from my first year as a graduate student I was overwhelmed by his patience and generosity, and he has continued to prove himself an exceptional mentor. His guidance throughout the years has been absolutely vital. I always enjoy our meetings and will miss them when I leave. I cannot express enough my immense gratitude. I feel very lucky to have had him as an advisor. I would also like to thank Professor Ettore Aldrovandi, Professor Kyounghee Kim, and Professor Kathleen Petersen. Their support throughout the writing of this dissertation is greatly appreciated. Professor Aldrovandi also taught me in several courses and I believe I benefited both as a student and as an educator by being his pupil. I owe much thanks to Professor Bernd Sturmfels and Professor Greg Smith. I learned so many valuable lessons in my time as an “apprentice” at the Fields Institute, and I am grateful for the experience. I have also grown from my experiences as a participant in the Macaulay2 boot camps under the supervision of Professor Mike Stillman and Professor Dan Grayson. As a young graduate student, my first time participating in a boot camp was an invigorating experience that fueled much of the work around this dissertation. I thank my friends and family for their continued love and support. I am lucky to have made many good friends as a graduate student, and they have made my journey all the better. Thanks especially to Ethan, Dan, Ivan, and Tony. I thank my best friend Stuart, who has been there for me since the beginning. I thank my aunt Trish for all her support. I thank my father Chris, my step-mother Rose, my step-father Tony, and my brother Nolan for believing in me and for all their encouragement. Finally, I thank my mom Lori; her unwavering support has meant the world to me. Lastly, I owe everything to my wife Debby. Thank you.

iii TABLE OF CONTENTS

ListofFigures ...... vi Abstract...... vii

1 Introduction 1 1.1 Segreclasses...... 1 1.1.1 Chern and Segre classes of vector bundles ...... 1 1.1.2 Segre class of a subvariety ...... 2 1.1.3 An algorithm for s(X,Y ) ...... 3 1.2 Monomial principalization ...... 3 1.3 Monomial planar Cremona transformations ...... 3 1.3.1 Cremona transformations ...... 3 1.4 Multiviewvariety...... 4 1.4.1 Euclidean distance degrees ...... 5 1.4.2 Chern-Matherclasses ...... 5 1.5 Tritangents to canonical space sextics ...... 5 1.6 Enriquessurfaces...... 6

2 Computing Segre classes in arbitrary projective varieties 7 2.1 Introduction...... 7 2.2 Background...... 8 2.2.1 Previous approaches ...... 9 2.3 Segre classes commute with intersecting by eﬀective Cartier divisors ...... 10 2.4 Degrees of Segre classes via linear projection ...... 12 2.5 An algorithm for computing Segre classes ...... 13 2.5.1 Setup ...... 13 2.5.2 Linear system of equations ...... 14 2.5.3 Algorithm ...... 15 2.6 Applications...... 15 2.6.1 Chernclasses ...... 16 2.6.2 Polar classes ...... 17 2.6.3 Euclidean distance degrees ...... 18 2.6.4 Degrees of projections of degeneracy loci ...... 18 2.6.5 Intersectionproducts...... 19

3 Monomial principalization in the singular setting 23 3.1 Introduction...... 23 3.2 Deﬁnitionsandexamples...... 23 3.3 R.c. monomial blowups preserve regular crossings ...... 25 3.4 Proof of main theorem ...... 29

iv 4 Classiﬁcation of the monomial Cremona transformations of the plane 32 4.1 Introduction...... 32 4.2 Some convex geometry ...... 33 4.2.1 Triangulations ...... 34 4.3 Computingmultidegreesviapolyhedra ...... 35 4.3.1 Multidegrees as volumes ...... 36 4.4 Newton polyhedra of Cremona transformations ...... 37 4.4.1 Main case ...... 37 4.4.2 Othercases ...... 39 4.5 Number of monomial Cremona transformations ...... 42 4.5.1 Similarity ...... 44

5 The Chern-Mather class of the multiview variety 45 5.1 Introduction...... 45 5.2 Deﬁnitions and notation ...... 48 5.3 A resolution of the multiview variety ...... 49 5.3.1 Constructing a resolution of φ ...... 50 5.3.2 The Chow ring of P˜3 ...... 51 5.3.3 The Chern class of the resolution ...... 53 5.4 Higherdiscriminants ...... 55 5.4.1 Higher discriminants of the resolution φ˜ ...... 56 5.5 The Chern-Mather class of the multiview variety ...... 58 5.5.1 Thebasicsetup...... 58

5.5.2 Calculating EuMVN (x)...... 58 5.5.3 The ED degree of the multiview variety ...... 60

6 Tritangent planes to space sextics: the algebraic and tropical stories 62 6.1 Introduction...... 62 6.2 Algebraic space sextics ...... 64 6.3 Tritangents to real space curves ...... 66 6.4 Tropicalspacesextics ...... 71 6.5 Tropical divisors, theta characteristics and tritangent planes ...... 74

7 Equations and tropicalization of Enriques surfaces 80 7.1 Introduction...... 80 7.2 Background ...... 81 7.3 Enriques surfaces via K3 complete intersections in P5 ...... 82

Appendix A Source code for Macaulay2 package FMPIntersectionTheory 87

Bibliography ...... 98 BiographicalSketch ...... 104

v LIST OF FIGURES

2.1 Diagram of blowups BlX Y and BlX K K ...... 11 ∩ 3.1 Examples...... 24

4.1 Schematic drawing of a Newton polyhedron ...... 35

4.2 Graph of ϕ ...... 36

4.3 Another Newton polyhedron ...... 38

4.4 Newton polyhedron for the standard involution ...... 40

4.5 Newton polyhedron when f =0 ...... 41

5.1 Schematic of three cameras ...... 49

6.1 Union of lines and a smooth cubic...... 67

6.2 Construction of sextic with ﬁve ovals ...... 67

6.3 Two (2, 1)-tritangents projected to xy-plane...... 69

6.4 The intersection of a cubic and a quadric yields a sextic...... 70

6.5 A real sextic curve with a single connected component...... 70

6.6 Atropicalellipticcurve...... 72

6.7 A tropical P1 P1 curve of bi-degree (1,2)...... 73 × 6.8 Twotropicalcurvesmeetingwithmultiplicity2...... 74

6.9 Tropical curves intersecting stably at two points...... 74

6.10 Two theta characteristics on a curve of genus 2...... 75

6.11 15 tritangents to a tropical sextic...... 78

vi ABSTRACT

This dissertation presents studies of effective methods in two main areas of algebraic geometry: intersection theory and characteristic classes, and combinatorial algebraic geometry. We begin in chapter 2 by giving an effective algorithm for computing Segre classes of subschemes of arbitrary projective varieties. The algorithm presented here comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space. To our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities. In chapter 3, we generalize an algorithm by Goward for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field, proving that the more general class of r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension 2 r.c. monomial centers. The main result of chapter 4 is a classification of the monomial Cremona transformations of the plane up to conjugation by certain linear transformations. In particular, an algorithm for enumerating all such maps is derived. In chapter 5, we study the multiview varieties and compute their Chern-Mather classes. As a corollary we derive a polynomial formula for their Euclidean distance degree, partially addressing a conjecture of Draisma et al. [35]. In chapter 6, we discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We investigate the situation for real and tropical sextics. In chapter 7, we explicitly compute equations of an Enriques surface via the involution on a K3 surface.

vii CHAPTER 1

INTRODUCTION

This work reports on results in algebraic geometry, particularly in intersection theory and combinatorial algebraic geometry, with a focus on formally effective methods. An effective method for a class of problems is an algorithm or procedure which is deterministic, finite, and guaranteed to produce the correct answer. Chapter 2 appeared as [48] and chapter 3 appeared as [47]. Chapter 6 and (an extended version of) chapter 7 will appear in the forthcoming [71]. In this introduction we give an extended preview of each chapter, summarizing the necessary prerequisites and main results.

1.1 Segre classes

In chapter 2, we give an algorithm for computing Segre classes of subschemes of arbitrary projective varieties by computing degrees of a sequence of linear projections. Based on the fact that Segre classes of projective varieties commute with intersections by general effective Cartier divisors, we can compile a system of linear equations which determine the coefficients for the Segre class pushed forward to projective space. The algorithm presented in section 2.5 comes after several others which solve the problem in special cases, where the ambient variety is for instance projective space; to our knowledge, this is the first algorithm to be able to compute Segre classes in projective varieties with arbitrary singularities.

1.1.1 Chern and Segre classes of vector bundles

To introduce Segre classes, we’ll ﬁrst go through Chern classes. Let X be a smooth variety over C, and consider a vector bundle E of rank r on X. The ith Chern class of E is a class c (E) A (X) i ∈ ∗ in the Chow ring of X. For this case, A∗(X) can be thought of as the even Borel-Moore cohomology H2 (X, Z). The class c (E) can be thought of as the degeneracy locus of r i + 1 general sections ∗ i − of E. For example if E has rank one, then c (E) [X] is the divisor on X corresponding to the 1 ∩ zeros and poles of a general section of E. Then we see that if E is the trivial bundle, ci(E) = 0 for

1 all i> 0 since general sections have no zeros or poles. For nontrivial E, we should expect that the

Chern classes ci(E) are nonzero for i = 0 ...r and are zero for all i>r. The total Chern class of E is 1+ c (E)+ + c (E) A (X). We can add an indeterminate 1 ··· r ∈ ∗ to this expression to track the grading, which yields the Chern polynomial

r i 2 ct(E) := ci(E)t =1+ c1(E)t + c2(E)t + .... Xi=0 Chern classes satisfy a few very useful properties, one of which is the Whitney sum formula: if

0 E F G 0 → → → → is an exact sequence of vector bundles on X, then the Chern classes are related by ct(F ) = ct(E)ct(G). Additionally, if f : Y X is a proper morphism, then f (c(f ∗(E)) α)= c(E) f (α) → ∗ ∩ ∩ ∗ ∈ A (X). ∗ We can now deﬁne the Segre classes as the coeﬃcients of the inverse power series of the Chern i 1 polynomial. That is, let i∞=0 si(E)t := (ct(E))− . Although the Chern classes of E vanish for i>rk(E), this is not theP case for the Segre classes of E.

1.1.2 Segre class of a subvariety

If V X is a smooth subvariety then we have an exact sequence involving tangent bundles ⊂

0 TV TX N X 0 → → |V → V → and the cokernel NV X is called the normal bundle to V in X. The Segre class of NV X is s(NV X)= (c(N X)) 1 = c(TV ) by the Whitney sum formula. V − c(TX V ) | Segre classes are more general than Chern classes, in that they are defined for “singular” vector bundles. In this context we get a normal cone C X for an arbitrary closed embedding V X, V ⊂ and the Segre class is called the Segre class of V in X and written s(CV X) =: s(V,X). A proper definition is given in section 2.2, but for now we will mention a property which ostensibly suffices to compute them. The Segre class of a subvariety is a birational invariant, that is, if f : X X is a proper birational morphism such that f 1(V ) = V X , then ′ → − ′ ⊂ ′ f s(V ′,X′) = deg(f)s(V,X). Thus, over C, we can always compute the Segre class of V in X by ∗ replacing them with a pair V X for which the normal bundle exists. ′ ⊂ ′

2 1.1.3 An algorithm for s(X,Y )

There have been several contributions [3,38,50,51,62] to the eﬀort of computing Segre classes, giving algorithms for s(X,Y ) when Y is either projective space or a smooth toric variety. The algorithm presented in section 2.5 is unique in that it can be applied when Y is any projective variety.

1.2 Monomial principalization

Let Y be a variety and X Y a subvariety defined by a sheaf of ideals . A princi- ⊂ IX ⊂ OY palization of is a birational map π : Y 99K Y such that Y is a smooth variety and the inverse IX ′ ′ image sheaf ′ is locally principal. IX OY The principalization of an ideal sheaf on a variety is an important process in the theory of resolution of singularities. However, it is also related to Segre classes, since π s(π∗(X),Y ′) = ∗ s(X,Y ). In chapter 3, we generalize an algorithm by Goward [45] for principalization of monomial ideals in nonsingular varieties to work on any scheme of finite type over a field. The normal crossings condition considered by Goward is weakened to the condition that components of the generating divisors meet as complete intersections. This leads to a substantial generalization of the notion of monomial scheme; we call the resulting schemes regular crossings (r.c.) monomial. We prove that r.c. monomial subschemes in arbitrarily singular varieties can be principalized by a sequence of blow-ups at codimension 2 r.c. monomial centers.

1.3 Monomial planar Cremona transformations

The main result of the chapter 4 is a classiﬁcation of the monomial Cremona transformations of the plane up to conjugation by certain linear transformations. In particular, the monomial Cremona transformations can be enumerated, and up to conjugation there are φ(d) transformations of degree d 3, where φ is Euler’s totient function. ≥ 1.3.1 Cremona transformations

Let U, V X be dense aﬃne open subsets. We say morphisms f : U Y and g : V Y are ⊂ → → equivalent if they agree on the overlap U V . This deﬁnes an equivalence relation on the set of ∩

3 morphisms from dense open subsets of X to Y . An equivalence class ϕ under this relation is called a rational map from X to Y and is denoted ϕ : X 99K Y . Given X,Y Pn, we think of rational maps ϕ : X 99K Y as rules x (f (x) : : f (x)) ⊂ 7→ 0 ··· n which are only defined on Y V (f ,...,f ). The scheme defined by (f ,...,f ) is called the base \ 0 n 0 n scheme of ϕ and the underlying set is the base locus. For example the rational map P2 99K P1 given by (x : y : z) (x2 : y2) is defined everywhere except its base locus (0 : 0 : 1) P2. 7→ ∈ Given ϕ : X 99K Y , if there exists ρ : Y 99K X such that ϕ ρ = id and ρ ϕ = id , then we ◦ X ◦ Y say ϕ is birational. A birational map Pn 99K Pn is known as a Cremona transformation. The most famous Cremona transformation is Cremona’s involution of the plane: (x : y : z) (yz : xz : xy). 7→ A special feature of this map is that it is defined by monomials. In chapter 4, we determine all possible monomial Cremona transformations of P2 in the following way. Recall that a totative of n N is an integer 0 < k n which is coprime to n. Given a ∈ ≤ degree δ and a totative k of δ, there is a unique monomial planar Cremona transformation. For example, if δ = 15, its totatives are 1, 2, 4, 7, 8, 11, 13, 14 . The monomial Cremona transformation { } corresponding to 11 is (x : y : z) (z15 : x11y4 : x8y3z3). 7→ Up to swapping variables in the domain and codomain, there is a 1-1 correspondence between monomial Cremona transformations “of degree δ” and totatives of δ.

1.4 Multiview variety

The multiview variety associated to a collection of N cameras records which sequences of image points in P2N can be obtained by taking pictures of a given world point x P3 with the cameras. ∈ In order to reconstruct a scene from its picture under the different cameras it is important to be able to find the critical points of the function which measures the distance between a general point u P2N and the multiview variety. In chapter 5, we calculate a specific degree 3 polynomial that ∈ computes the number of critical points as a function of N. In order to do this, we construct a resolution of the multiview variety, and use it to compute its Chern-Mather class.

4 1.4.1 Euclidean distance degrees

Given an affine variety X An and a point u An we define the squared Euclidean dis- ⊂ ∈ tance function on X to be d (x) = n (x u )2. The Euclidean distance degree of X, denoted u i=1 i − i EDD(X), is the number of critical pointsP of du(x) on X. This number is constant for u in a dense open subset of An. For a projective variety, the Euclidean distance degree is the Euclidean distance degree of its affine cone.

1.4.2 Chern-Mather classes

For a smooth variety X, the Chern class of X is c(TX ) [X] A (X), the total Chern class ∩ ∈ ∗ of its tangent bundle. However, when X is not smooth, so that it doesn’t have a tangent bundle, there are several alternative Chern classes that agree with this one in the smooth case. One such option is the Chern-Mather class, which is defined in terms of the Nash blowup of ν : X˜ X. The → Nash blowup of Xk Pn is the closure of the graph of the Gauss map X 99K Gr(k,n) sending each ⊂ smooth point to its tangent plane. The Nash blowup comes equipped with the tautological Nash tangent bundle T˜, and we define the Chern-Mather class of X by cMa(X)= ν (c(T˜) [X˜]). ∗ ∩ For our purposes, the key is that the Chern-Mather class of X Pn carries the same information ⊂ as the class of the conormal variety in A (Pn Pn). In nice situations (like in chapter 5), the ∗ × Euclidean distance degree of a projective variety X is the sum of the coefficients in the class of the conormal variety and therefore can be computed from the Chern-Mather class. It was conjectured in [35] that the ED degree of the affine multiview variety is a cubic polynomial in the number of cameras. The main result of this chapter is that the ED degree is at least bounded by a cubic polynomial.

1.5 Tritangents to canonical space sextics

In chapter 6, we discuss the classical problem of counting planes tangent to general canonical sextic curves at three points. We determine the number of real tritangents when such a curve is real. We then revisit a curve constructed by Emch with the greatest known number of real tritangents, and conversely construct a curve with very few real tritangents. Using recent results on the relation between algebraic and tropical theta characteristics, we show that the tropicalization of a canonical sextic curve has 15 tritangent planes.

5 1.6 Enriques surfaces

Enriques surfaces are surfaces with irregularity q=0 and non-trivial canonical bundle whose square is trivial. Enriques gave the ﬁrst example of such a surface to show that an algebraic surface with irregularity q = 0 and genus pg = 0 need not be rational. In chapter 7, we explicitly compute equations of an Enriques surface via the involution on a K3 surface. It is a bit surprising that explicit equations do not seem to be recorded in the literature. The equations can be used to compute tropicalizations, for instance, and this is done in the forthcoming [24], where we explore the tropicalization of an Enriques surface.

6 CHAPTER 2

COMPUTING SEGRE CLASSES IN ARBITRARY PROJECTIVE VARIETIES

2.1 Introduction

Segre classes are of fundamental importance in Fulton-MacPherson intersection theory, and there has been a growing recent interest in their concrete computation. For one thing, the Segre class of the singularity subscheme of a hypersurface in PN can be used to compute its Chern-

Schwartz-MacPherson (cSM ) class, and therefore also its topological Euler characteristic. It also features heavily in enumerative problems (see [42, 9.1] for examples). § For the Segre class of a closed subscheme i : X ֒ PN of projective space, [3] gave the first → algorithm for the computation of i s(X, PN ) A (PN ). Following this, a new algorithm (call it ∗ ∈ ∗ EJP) was contributed by [38]. A key feature of EJP is that it could be implemented using techniques from numerical algebraic geometry. Following these initial developments, several improvements have been made. First, [62] extended the EJP algorithm to the computation of j s(X,T ) ∗ ∈ A (T ) where j : X ֒ T is a closed embedding in a smooth projective toric variety (with some ∗ → minor restrictions on T ). The current state-of-the-art in this direction is the algorithm of [50, 51] which offers significant performance improvements over previous algorithms for both i s(X, PN ) ∗ and j s(X,T ). ∗ In a different direction, Aluffi [4] found a combinatorial method for computing s(X,Y ) where Y is a variety (over an algebraically closed field) and X Y is an r.c. monomial subscheme. ⊂ This places a strong restriction on the possible subschemes that can be considered, but allows the most freedom for the ambient scheme. However, this method requires computations of intersection numbers of divisors on Y and so requires more input than just the ideals of X and Y . Despite the range of results just mentioned, it is not difficult to find simple situations in which none of these methods apply. For instance, if C = Proj k[x,y,z,t]/(xy z2) is the singular cone − and L is the subvariety cut out by the ideal (¯x, z¯), then none of the aforementioned methods can be used to compute s(L,C), whereas the algorithm of the present chapter puts no restrictions on

7 the ambient variety, so it could be used here.

The main result of this chapter is algorithm 1 for computing i s(X,Y ) A (PN ) with Y PN ∗ ∈ ∗ ⊂ a variety and X ֒ Y an arbitrary closed subscheme. The algorithm is based on the fact that, under → suitable hypotheses, Segre classes commute with taking hypersurface sections (Corollary 2.3) and a projection technique which produces a system of linear equations in the coefficients of i s(X,Y ) ∗ (proposition 2.4). Section 2.2 begins with the basic setup and assumptions for the rest of the chapter. In section 2.2.1 the algorithms of [3] and [38] are reviewed and their relation to the present work is indicated. In section 2.3 we study the effect on Segre classes when intersecting by divisors, and section 2.4 is devoted to the proof of proposition 2.4, which yields a linear equation satisfied by the coefficients of the Segre class. Finally, in section 2.5 the main algorithm for the computation of Segre classes is presented. Then in section 2.6 several applications of the algorithm are discussed, including computations of the Chern-Schwartz-MacPherson classes, Chern-Mather classes, and polar classes of projective varieties. We also show how to use algorithm 1 to compute Euclidean distance degrees. Finally, we describe some situations where the Segre class is enough to compute intersection products. Examples of each of these are worked out in detail.

2.2 Background

Let Xr Y n PN be closed embeddings of schemes over a ﬁeld k. Further assume that Y is ⊂ ⊂ a variety, or more generally a reduced subscheme of pure dimension n. Given the blowup square

E Y˜ π′ π X Y the Segre class of X ( Y may be deﬁned as

[E] s(X,Y )= π′ s(E, Y˜ )= π′ ( ⌢ [Y˜ ]), ∗ ∗ 1+ E a class in A (X). In general, it will not be possible to compute s(X,Y ), as the Chow group for X ∗ may not be known, but if we are willing to settle for the pushforward to PN , then the class can be

8 0 1 r N N written i s(X,Y )= s0[P ]+ s1[P ]+ + sr[P ] A (P ), where i : X ֒ P is the inclusion. In ∗ ··· ∈ ∗ → this chapter, we exhibit an algorithm for computing these si. Note: the setup just presented will remain ﬁxed throughout the chapter. Any symbols used above will be used exactly this way from here on.

2.2.1 Previous approaches

We now summarize some approaches that have previously been used in the computation of Segre classes. Let X Y PN be closed embeddings with X deﬁned by equations f ,...,f all ⊂ ⊂ 0 k k of degree deg(fi)= d. We can deﬁne a rational map pr : Y 99K P

Y˜ π ˜pr

pr Y Pk

by pr(p)=[f (p): : f (p)] and resolve the indeterminacies by lifting to Y˜ , the blowup of Y 0 ··· k along the base scheme X. Aluffi [3] gave a procedure for computing the Segre class i (X, PN ) for a closed subscheme ∗ X PN . The procedure is founded on the observation that the pushforward s(X, PN ) is determined ⊂ by the class of the shadow of the blowup of PN along X. This class is G = a [P0]+ + a [PN ] 0 ··· N where N i ai = π (c1( ˜pr∗ (1)) − ⌢ [Y˜ ]) ∗ O Z are the projective degrees of the map pr. Once these coefficients have been found, the Segre class can be computed as N 1 i s(X, P ) = 1 c( (d))− ⌢ (G (d)), ∗ − O ⊗O see [3, Theorem 2.1]. Helmer [50] has recently expanded on this approach by giving a new way to compute the projective degrees ai. Eklund et al. [38] gave a different procedure (call it EJP) for computing i s(X, PN ) based on ∗ computing a residual intersection. If g1,...,gN r are polynomials of degree d which vanish on − X, then we can consider the linear system generated by the corresponding sections in H0( IX ⊗ (d), PN ). The base locus of this linear system contains X and so we may write the intersection O product of the corresponding hypersurfaces as a part contributed by X along with a “residual”

9 part: N D1 ... DN r = c( ) ⌢ s(X, P ) N r + R0 · · − { N } − The method of EJP is to sequentially add polynomials to get g ,...,g for N r m N, getting 1 m − ≤ ≤ residual contributions R0,..., Rr. They then use the formula [38, Theorem 3.2]

p m p i m d − sr i = d deg(Rp), p i − − Xi=0 − where p = m (N r), to write the coeﬃcients of i s(X, PN ) in terms of the degrees of these − − ∗ residual intersection classes.

From the equations of EJP one can construct the following linear system for the si in terms of N,d,r and the residuals:

1 N d N d2 ... N dr s dN deg(R ) 1 2 r 0 − r N 1 N 1 r 1 N 1 0 1 − d ... − d −   s1  d − deg(Rr 1) 1 r 1 − − − . . . . .. .. .   .  =  .  (2.1) . . . . .   .   .   N r+1     N r+1  0 0 ... 1 − d  sr 1 d − deg(R1)  1   −   −       N r  0 0 ... 0 1   sr   d − deg(R0)       −        Algorithm 1 can be seen as a generalization of EJP in the sense that the system of equations produced using algorithm 1 exactly matches eq. (2.1) when Y = PN . Indeed, it was observed by [50, Section 3.2] that the projective degrees of pr actually correspond with the residual degrees, that is, aN r+j = deg(Rj) for 0 j r, and performing this substitution produces eq. (2.3) on − ≤ ≤ page 14. However, the geometric pictures in the two algorithms are diﬀerent, even in the case Y = PN , since EJP works by constructing a system of equations which constrain s(X,Y ) and algorithm 1 works by producing several Segre classes and comparing them.

2.3 Segre classes commute with intersecting by eﬀective Cartier divisors

The main goal of this section is theorem 2.2. This result can be found in many places, for instance [10, Lemma 4.1], but a detailed proof was harder to ﬁnd so we give one here. For this section, we will ﬁrst need to review the concept of distinguished varieties. Given a closed embedding X Y , let C Y denote the normal cone to X in Y . Let C ,...,C be the ⊂ X 1 l

10 irreducible components of CX Y and let Zi denote the support of Ci in X. These Zi are called the distinguished varieties for X in Y . It should be noted that the Zi are not necessarily distinct; two irreducible components Ci,Cj may have the same support Zi = Zj. Now let K Y be a Cartier divisor and let K˜ be the blowup of K along X K, with exceptional ⊂ ∩ divisor F . Then F = E K˜ and we have the commutative diagram in ﬁg. 2.1 where the front and ∩ back faces are blowup squares.

E Y˜

π′ π F K˜

ρ′ ρ X Y

X K K ∩

Figure 2.1: Diagram of blowups BlX Y and BlX K K ∩

The next proposition says that if K is in general position with respect to the distinguished subvarieties of X in Y , then the proper transform of K in the blowup along X has no extra components.

Proposition 2.1. Let K Y be an eﬀective Cartier divisor. If K does not contain any of the ⊂ distinguished varieties for X Y , then the proper transform K˜ is equal to the total transform ⊂ 1 π− (K).

1 Proof. It suﬃces to show that each irreducible component of π− (K) is contained in K˜ . Aside from those in K˜ , the irreducible components of π 1(K) each lie in π 1(K Z) for some distinguished − − ∩ variety Z. But K Z is an eﬀective Cartier divisor on Z, so its pullback is one on π 1(Z) and ∩ − 1 must have codimension at least 2 in Y˜ . Thus since π− (K) is pure of codimension 1 in Y˜ , there is no component of π 1(K) contained in π 1(K Z). − − ∩ We can now prove the formula below which shows how the Segre classes s(X,Y ) and s(X K,K) ∩ are related for divisors K Y which do not contain the distinguished varieties of X. ⊂

11 Theorem 2.2. For an eﬀective Cartier divisor K Y which does not contain any of the distin- ⊂ guished varieties for X in Y ,

K s(X,Y )=s(X K,K) A (X K). · ∩ ∈ ∗ ∩ E ˜ E ˜ Proof. By deﬁnition, we have s(X,Y )= π′ 1+E ⌢ [Y ] and s(X K,K)= ρ′ 1+E ⌢ [K] . We ∗ ∩ ∗ arrive at the desired equation by combining proposition 2.1 with the projection formula for divisors and commutativity of Chern classes: E K s(X,Y )= K π′ ⌢ [Y˜ ] · · ∗ 1+ E E = ρ′ π′∗(K) ⌢ [Y˜ ] 1+ E ∗ E = ρ′ ⌢ [K˜ ] 1+ E ∗ = s(X K,K). ∩ We now give the precise statement that will be needed for algorithm 1.

Corollary 2.3. For a general hypersurface H PN of any ﬁxed degree, ⊂ H s(X,Y )=s(X H,Y H) A (X H). · ∩ ∩ ∈ ∗ ∩ Proof. Bertini’s Theorem says that for general H PN , the divisor H Y is eﬀective and does not ⊂ ∩ contain any distinguished varieties for X Y , so theorem 2.2 applies. ⊂ 2.4 Degrees of Segre classes via linear projection

Now assume X is the base scheme of a (k + 1 dimensional) linear system L H0(Y, (d)) and ⊂ O let pr : Y 99K Pk be the associated rational map. As usual, denote by π : Y˜ Y the blowup of Y → along X and let ˜pr : Y˜ Pk be the lift of the projection map to Y˜ . In this setup, [42, Proposition → 4.4] gives the formula

c( (d))n ⌢ s(X,Y )= c ( (d))n deg(Y/˜ ˜pr(Y˜ )) c ( (1))n ⌢ [ ˜pr(Y˜ )]. O 1 O − 1 O XZ YZ PZk N 0 1 r If the pushforward of s(X,Y ) to P is i s(X,Y )= s0[P ]+ s1[P ]+ + sr[P ] then the left-most ∗ ··· degree is r N 1 n 0 1 r n i (1 + d[P − ]) (s [P ]+ s [P ]+ + s [P ]) = d s (2.2) 0 1 ··· r 0 i i i=0 X

12 while for the middle term we have c ( (d))n = dn deg(Y ). Y 1 O As for the last term, we can interpretR the factors as follows. The degree deg(Y/˜ ˜pr(Y˜ )) is the degree of the map ˜pr when this is generically ﬁnite, and is 0 otherwise, while c ( (1))n ⌢ [ ˜pr(Y˜ )] Pk 1 O is the degree of ˜pr(Y˜ ) Pk if this subvariety has dimension n = dim Y andR is 0 otherwise. Since ⊂ the second term will yield 0 exactly when the ﬁrst does, we can consolidate these by writing

n deg(Y/˜ ˜pr(Y˜ )) c1( (1)) ⌢ [ ˜pr(Y˜ )] = deg( ˜pr) deg( ˜pr(Y˜ )) · k O ZP with the understanding that the deg( ˜pr) should be 0 if the map is not generically ﬁnite. Altogether, this lets us write dim X n dis = dn deg(Y ) deg( ˜pr) deg( ˜pr(Y˜ )) i i − Xi=0 Since the map ˜pr factors as pr π and π is an isomorphism away from X, we see that deg( ˜pr) = ◦ deg(pr) and ˜pr(Y˜ )= ˜pr(Y˜ E)= pr(Y X). This allows us to completely remove consideration − − of Y˜ in the computation of s(X,Y ). We collect these observations in the following proposition.

Proposition 2.4. In the setup of this section, dim X n dis = dn deg(Y ) deg(pr) deg(pr(Y X)). i i − − Xi=0 This should be seen as a linear equation in the unknowns s0,...,sdim X .

2.5 An algorithm for computing Segre classes

We can ﬁnally give the main formula in proposition 2.4 and describe the algorithm for computing the Segre class i s(X,Y ) A (PN ). ∗ ∈ ∗ 2.5.1 Setup

Start by setting X0 = X and Y0 = Y and assume that X0 is cut out in Y0 by homogeneous equations f1,...,fm. If d = maxi(deg(fi)), we can consider X0 as the base locus of a linear system in H0(Y , (d)) by “padding” the f appropriately. That is, if deg(f ) < d we replace f by the 0 O i i i polynomials gi,1,...,gi,q where gi,j = fiσj and σ1,...,σq are the standard monomials of degree d deg(f ) in the coordinates of PN . Relabeling, write the set of g for all i,j as g ,...,g . Then − i i,j 0 k X0 is the base locus of the linear system generated by the sections corresponding to these gi.

13 2.5.2 Linear system of equations

To shorten the statements that follow, deﬁne

δ = ddim Yj deg(Y ) deg(pr )deg(pr (Y X )), j j − j j j − j 99K k where pr0 : Y0 P is the projection defined previously and prj will be defined analogously for 0 1 r objects Xj Yj. Writing i s(X0,Y0)= s0[P ]+ s1[P ]+ + sr[P ] as before, we get ⊆ ∗ ··· r n dis = δ i i 0 Xi=0 by proposition 2.4. Now choose a general hyperplane section H Y àla Corollary 2.3. If we set Y = Y H ⊂ 0 1 0 ∩ and X = X H we can repeat the procedure above, with the sections of the new linear system 1 0 ∩ given by restricting g0,...,gk to Y1. This gives a new projection pr1. By Corollary 2.3, we have H s(X ,Y )= s(X ,Y ) so the resulting linear equation will be · 0 0 1 1 r n 1 i 1 − d − s = δ . i 1 i 1 Xi=1 − Repeating this process of taking hyperplane sections will yield a system of equations A~s = ~δ. Explicitly, n n 2 n r 1 1 d 2 d ... r d s0 δ0 n 1 n 1 r 1 0 1 −1 d ... r−1d −   s1   δ1  − . . . . . . . .. .. .   .  =  .  (2.3) . . . . .   .   .        0 0 ... 1 n r+1 d  s  δ   −1   r 1  r 1    −   −  0 0 ... 0 1   s   δ     r   r  1       and solving for A− gives 1 n ( d) n ( d)2 ... n ( d)r δ s 1 − 2 − r − 0 0 n 1 n 1 r 1 0 1 −1 ( d) ... r−1( d) −   δ1   s1  − − − . . . . . . . .. .. .   .  =  .  . (2.4) . . . . .   .   .        0 0 ... 1 n r+1 ( d)  δ  s   −1   r 1  r 1  −   −   −  0 0 ... 0 1   δ   s     r   r        From this we get the following closed form expression for the si.

Proposition 2.5. With notation as in this section, r n i j i s = − ( d) − δ . i j i − j Xj=0 −

14 2.5.3 Algorithm

The algorithm can now be summarized quite concisely. Given X Y PN , calculate 0 ⊂ 0 ⊂ deg(Y ), deg(pr ) and deg(pr (Y X )). Replace X and Y with X := X H and Y := Y H 0 0 0 0 − 0 i i i+1 ∩ i+1 ∩ for a suitable hyperplane H PN and repeat the calculations. Continue until dim X = 0. For ⊂ i each i, calculate si using proposition 2.5. As a more verbose presentation we oﬀer the pseudocode version of the algorithm for computing s(X,Y ). In practice, the inputs are two ideals, I ,I k[x ,...,x ] deﬁning the schemes X,Y X Y ⊂ 0 N ⊂ N Pk respectively.

Algorithm 1: Computation of s(X,Y) Input: a variety Y n PN and closed embedding Xr Y given by a linear system in (d) ⊂ → O Output: the Segre class s(X,Y ) pushed forward to PN

X X, Y Y 0 ← 0 ← for i = 0 to r do δ dn deg(Y ) deg(pr )deg(pr (Y X )) i ← i − i i i − i Choose generic hyperplane H PN as in Corollary 2.3 ⊂ Y Y H, X X H i+1 ← i ∩ i+1 ← i ∩ end for i = 0 to r do r n i j i si j=0 j −i ( d) − δj ← − − end P return s hN + + s hN r 0 ··· r −

The algorithm has been implemented in Macaulay2 and an implementation should soon be available for SageMath as well. The Macaulay2 package interfaces with Schubert2, as can be seen in the examples to follow.

2.6 Applications

In this section we give examples of the general types of computations that can be made if Segre classes are accessible. Along with Chern classes (like that of Mather, Schwartz-MacPherson) one can also compute polar classes, which can be used to compute Euclidean distance degrees. We also

15 discuss how Segre classes can be applied in the computation of degrees of projections of degeneracy loci. Finally, we give several examples of intersection product computations, the ﬂagship application of the Segre class.

2.6.1 Chern classes

Let Z be a hypersurface of a nonsingular m-dimensional variety M and let J be the singularity subscheme of Z, i.e., the subscheme deﬁned by the Jacobian ideal of Z. Several important objects instrinsic to Z can be computed if the Segre class of J in Z or M is known. [2, Lemma I.4] gave a formula for the Chern-Schwartz-MacPherson class of Z in terms of s(J, M): [Z] c (Z)= c(TM) ⌢ +(c( (Z) ) ⌢ s(J, M))∨ (Z) . SM 1+ Z O |J ⊗O |J Similarly, the Chern-Mather class of Z can be written [9, Prop. 2.2]

[Z] c (Z)= c(TM) ⌢ + s(J, Z)∨ (Z) . Ma 1+ Z ⊗O |J where now the key input is s(J, Z). Both of these formulas are quite explicit, and can be implemented in [46] by a literal translation of the formulas above. For instance, Schubert2 provides the structure sheaf O and also provides adams which performs the ( ) operation, and our code · ∨ implements as **. Then if cZ is the cycle class of Z in A (PN ), the Chern-Mather class can be ⊗ ∗ computed with s = segreClass(iJ,iZ) chern(TM)*(cZ*(1+cZ)^(-1)+(adams(-1,s)**O(cZ)))

For example, we can compute c (X) and c (X) for X P3 the singular cone: SM Ma ⊂ i1 : PP3 = QQ[x,y,z,t] i2 : X = ideal x*y-z^2 i3 : chernMather(X) o3=2H +4H +2H 2,1 2,2 2,3 i4 : chernSchwartzMacPherson(X) o4=2H +4H +3H 2,1 2,2 2,3

The output of these two commands are elements of the IntersectionRing of P3. In this case they 2 1 0 2 1 0 say cSM (X) = 2[P ] + 4[P ] + 3[P ] and cMa(X) = 2[P ] + 4[P ] + 2[P ].

16 2.6.2 Polar classes

N N Now assume Z is a hypersurface in P and let Λk 1 P be a general (k 1)-dimensional − ⊂ − linear subspace. We deﬁne the kth polar locus of Z (relative to Λk 1) by −

Pk = x Zsm Λk 1 TxX . { ∈ | − ⊂ }

The kth polar class ̺k is the degree ̺k = [Pk]. [66, Theorem 2.3] showed that when Z is a hypersurface of degree d, the class [Pk] AN R1 k(Z) can be expressed in terms of the Segre class ∈ − −

0 N 1 s(J, Z)= s0[P ]+ + sN 1[P − ] ··· − of the singularity subscheme:

k 1 k k − k i i i N [Pk]=(d 1) h ⌢ [Z] (d 1) h ⌢si[P ] AN 1 k(P ). − − i − ∈ − − Xi=0 The polar classes ̺0, . . . , ̺N 1 are the degrees of the classes [Pk], so when Z is a hypersurface the − above formula gives

k 1 − k ̺ = [P ]= d(d 1)k (d 1)is . (2.5) k k − − i − i Z Xi=0 N Polar classes are projective invariants [52, 66]. If Z has codimension c > 1 in P and prΛ : PN 99K PN c+1 is the projection from a general (c 1)-dimensional linear subspace Λ PN then − − ⊂ ̺k(Z)= ̺k(prΛ(Z)) for all k. Thus algorithm 1 allows the computation of the polar classes of any projective variety. Moreover, [67, p. 19] showed that the polar classes carry the same information as the Chern- k Mather class. In particular, if ςk is the coeﬃcient of [P ] in cMa(Z),

k k n + 1 i n + 1 i ς = ( 1)i − ̺ and ̺ = ( 1)i − ς . k − k i k k − k i k Xi=0 − Xi=0 − An implication of this is that Chern-Mather classes are also projective invariants, and indeed Piene observed that if pr is a general linear projection as above,

cMa(pr(Z)) = pr cMa(Z) ∗ in A (pr(Z)). ∗

17 2.6.3 Euclidean distance degrees

For any subvariety Z M, an alternate but equivalent characterization of its polar classes is ⊂ the multidegree of the conormal cycle of Z in M. That is, let

N N ◦ = (p,h) P (P )∨ p Z and h is tangent to X at p . NZ ∈ × | ∈ sm The closure = PN (PN ) is the conormal variety of Z. It is irreducible of pure NZ NZ◦ ⊂ × ∨ N N dimension N 1 and its class in the Chow group A (P (P )∨) can be written − ∗ ×

N N 1 2 N [ Z ]= ̺0a b + ̺1a − b + + ̺N 1ab N ··· −

N N where a,b are the pullbacks of the hyperplane classes on P and (P )∨ respectively. For a general point y PN define d (x)= N (y x )2 to be the squared Euclidean distance ∈ y i=0 i − i from y to x. The Euclidean distance degree EDdeg(P Z) of an algebraic variety Z PN is the number ⊂ of critical points for dy restricted to Zsm. In [35, Theorem 5.4], the authors show that if does not meet the diagonal in PN (PN ) , NZ × ∨ then EDdeg(Z) is the sum of the polar classes of Z. For example ([35, Example 5.7]), consider Cayley’s quartic surface in C P3 defined by x3 xy2 xz2 + 2yzt xt2 = 0. The conormal ⊂ − − − variety is defined by 18 equations in P3 P3, whereas the Segre class of the singularity subscheme × is quite simple: i1 : C = ideal "x3-xy2-xz2+2yzt-xt2" i2 : J = ideal singularLocus C i3 : segreClass(J,C) o3 = 8H 2,3 so s(J, C) = 8[P0] and thus by eq. (2.5), the Euclidean distance degree is 3 + 6 + 4 = 13 (cf. [35]).

2.6.4 Degrees of projections of degeneracy loci

2 Let Pn 1 be the space of n n complex matrices. Let τ denote the locus of matrices of corank − × k n2 1 at least k. [5] considers the following problem: if S is a set of s coordinates on P − , and πS is n2 s 1 the projection from center V (S), what is the degree of the closure of πS(τk) in P − − ? An excess intersection computation yields the general solution (cf. [42, Proposition 4.4]):

18 n2 1 Lemma 2.6 ([5, Lemma 2.7]). Let H be the class of a hyperplane in P − . Then the degree d of the closure of πS(τk) is given by

k 1 n+i − k n2 k2 1 d = (1 + H) − − ⌢ s(L τ ,τ ). k+i − S ∩ k k i=0 k Y Z These Segre classes are great examples of problems for which a general computational method was not previously available. As an example of the efficacy of algorithm 1, the table below shows some computations for τ P8, the locus of 3 3 matrices of rank 2. The tableaux denote the 2 ⊂ × ≤ sets S, where the top left square is the coordinate of the first row and first column of the general matrix.

S i s(LS τ2,τ2) ∗ ∩ 25[P0] + 10[P1] 2[P2] [P3]+[P4] − − − 10[P0] 2[P2]+ [P3] − 3[P0] 3[P1]+ [P2] − 3[P0] 3[P1]+ [P2] − 11[P0] + 2[P1] − As an example we compute the ﬁrst row in the table: i1 : PP8 = QQ[v_0..v_8] i2 : I = minors(3, matrix{{v_0..v_2}, {v_3..v_5}, {v_6..v_8}}) i3 : J = ideal (v_0..v_3) i4 : segreClass(J,I) o4=H -H -2H +10H -25H 2,4 2,5 2,6 2,7 2,8

2.6.5 Intersection products

A basic intersection product setup in [42, Ch. 6] is a ﬁber square

j W V g f Xi Y

19 in which i is a regular embedding and V is a variety of dimension k. In this situation, we let N denote the pullback g∗NX Y of the normal bundle to X in Y and deﬁne the intersection product of X and V in Y to be

X Y V = c(N) ⌢ s(W, V ) k+r n . · { } − Assuming that f,i are really inclusions with X,V Y PN , we potentially have everything ⊂ ⊂ η needed to compute pushforward of X Y to PN . That is, if W ֒ PN is the inclusion and we can · → N ﬁnd a bundle Nˆ on P such that N = η∗Nˆ, then we can use the projection formula

η (X V )= η c(N) ⌢ s(W, V ) k+r n ∗ · ∗ { } − = c(Nˆ) ⌢η s(W, V ) . ∗ k+r n n o − Such is the situation when X PN and Y PN are both complete intersections. Then the ⊂ ⊂ N N normal bundle NX Y is isomorphic to the quotient NX P /i∗NY P and so its total chern class in PN can be written 4 1+ di[P ] i . (1 + e [P4]) Qj j Grassmannian of lines in P3. AsQ an example, consider G(2, 4) P5, the Grassmannian ⊂ of lines in P3, defined by the equation ab cd + ef = 0. Let Σ be the Schubert cycle of lines − 1 meeting a fixed line, with representative defined by the equation b = 0 on G(2, 4), and let Σ2,1 be the Schubert cycle of lines containing a fixed point and lying in a fixed plane, with representative defined by the equations b = d = e = f = 0. Say we wish to compute the pushforward to PN of the intersection product Σ Σ in G(2, 4). By the remarks above, we can write the chern class of 2,1 · 1 the normal bundle as 1+[P4] 4 NΣ2,1 G(2, 4) = 4 (1 + 2[P ]) and the other needed ingredient is s(Σ Σ , Σ ) = s(Σ , Σ ). In Macaulay2, we compute this 1 ∩ 2,1 1 2,1 1 by i1 : PP5 = QQ[a,b,c,d,e,f] i2 : G = ideal "ab-cd+ef" i3 : S1 = G + ideal b i4 : S21 = G + ideal (b,c,d,e) i5 : segreClass(S21,S1) o5 = H - H 2,4 2,5

20 which returns s(Σ , Σ )= [P0]+[P1]. Thus, the (pushed-forward) intersection product is 2,1 1 − (1+[P4])4 Σ Σ = ( [P0]+[P1]) 2,1 · 1 (1 + 2[P4]) − 0 = 2[P0]+[P1] + 2[P3] + 2[P4]+[P5] [P0]+[P1] − − 0 0 1 = [P ]+[P ] 0 =[P0] as expected since σ2,1σ1 = σ2,2 in A (G(2, 4)). ∗ Now let V be the subvariety of G(2, 4) deﬁned by b2 cf = 0. Then dim V = 3 and dim V = 2. − sing If we wish to compute the pushforward of Σ V we can use 1 · 1+[P4] 1 + 2[P4] c(N G(2, 4)) ⌢ s(Σ V,V ) = ⌢ s(Σ V,V ) { Σ1 1 ∩ }2 1 + 2[P4] 1 ∩ ( )2 = 1+[P4] 4[P0] 4[P1] + 4[P2] − 2 = 4[P2].

But this example is too easy, in the sense that it could have been computed as a product in P5. That is, since Σ is a complete intersection in G(2, 4), we could write Σ V as G(2, 4) H V 1 1 · · · where H is the hypersurface in P5 deﬁned by b = 0. Then Bezout’s Theorem says the intersection product has degree 4, so counting dimensions yields the result. In light of this we consider X G(2, 4) deﬁned by b = d = f = 0. This is still a complete ⊂ intersection in P5, but is not a complete intersection in G(2, 4), so the intersection product X V · cannot just be directly computed in P5. Notice also that X V , so the intersection is not proper. ⊂ The computation this time is

X V = c(N G(2, 4)) ⌢ s(X V,V ) · { X ∩ }1 1+[P4] 3 = ⌢ 2[P0]+[P1]+[P2] 1 + 2[P4] − ( )1 = 2[P1]

We can still verify this result by writing the product in Schubert cycles. Since V is a hypersurface in G(2, 4) of degree 4, it must be [V ] = 2σ1 A (G(2, 4)). With a little more work, one can check ∈ ∗ that [X]= σ1,1, and so we ﬁnd X V = 2σ σ = 2σ · 1 1,1 2,1

21 5 and Σ2,1 is a line in P .

22 CHAPTER 3

MONOMIAL PRINCIPALIZATION IN THE SINGULAR SETTING

3.1 Introduction

Monomial schemes are schemes defined as intersections of collections of components from a fixed normal crossing divisor in a nonsingular variety. In [Gow05], Goward proves that monomial schemes may be principalized by a sequence of blow-ups along codimension 2 monomial schemes. In the singular setting, this definition of monomial scheme is not available, because the notion of normal crossings requires a nonsingular ambient variety. We consider a much weaker condition, which makes no nonsingularity assumption on the ambient variety: essentially, divisors meet with ‘regular crossings’ if they intersect along subschemes with the expected dimension. (For example, any two distinct irreducible curves on a smooth surface meet with regular crossings, regardless of whether they are nonsingular or meet transversally.) This leads to a generalization of the notion of monomial schemes, “r.c. monomial schemes”. See section 3.2 for formal definitions. We extend Goward’s result to the r.c. monomial case, showing that this much larger class of subschemes can be principalized via Goward’s procedure. Our result has been used in recent work on computations of Segre classes, cf. [7], Theorem 1.1.

3.2 Deﬁnitions and examples

Throughout, X will denote a scheme of ﬁnite type over an arbitrary ﬁeld. By regular sequence, we mean a sequence x ,...,x of elements in a ring R such that (x ,...,x ) R is a proper ideal 1 n 1 n ⊂ and, for each i, the image of xi in R/(x1,...,xi 1) is a non-zerodivisor, see [37, p. 243] −

Deﬁnition 3.1 (regular crossings). Let Y ,...,Y X be Cartier divisors. We say that Y ,...,Y 1 n ⊂ { 1 n} has regular crossings if for every subset A Y1,...,Yn and every point p Y AY , the local ⊂ { } ∈ ∩ ∈ equations y for the Y A form a regular sequence at p. i ∈

23 Note that the definition requires each Yi to be cut out locally by a non-zerodivisor, making Yi an effective Cartier divisor in X. Note also that the condition places no restrictions on X. The following definition will be used only in the introduction to compare concepts.

Deﬁnition 3.2 (simple normal crossings). Let Y ,...,Y X be Cartier divisors. We say that 1 n ⊂ Y ,...,Y has simple normal crossings if for every subset A Y ,...,Y , the intersection { 1 n} ⊂ { 1 n} Z = Y AY is nonsingular with codimX Z =#A. If D = aiYi, with ai 0, we say D is a simple ∩ ∈ ≥ normal crossings divisor or s.n.c. divisor. P

The s.n.c. condition on singletons requires each Yi to be nonsingular, and the condition on the empty set means X itself must be nonsingular.

2 Example 3.3. Let Ak = Spec k[x,y] and consider the curves Y1 deﬁned by y and Y2 deﬁned by y2 x3. These two curves do not meet with simple normal crossings because y2 x3 is not smooth − − in the intersection. So Y + Y is not an s.n.c. divisor, but Y ,Y does have regular crossings 1 2 { 1 2} since y2 x3 is sent to a non-zerodivisor in the integral domain k[x,y]/(y). Observe (Figure 3.1a) − that these curves do not meet transversally.

2 3 (a) Curves y, y x in the plane (b) Curves x, y on xy z2 − −

Figure 3.1: Examples

Example 3.4. Now consider the cone C = Spec k[x, y, z]/(xy z2) and the subschemes Y ,Y cut − x y out by the ideals (x) and (y), respectively. Since C is singular, there are no s.n.c. divisors in sight.

24 However (k[x, y, z]/(xy z2))/(y) = k[x, z]/(z2), so y,x forms a regular sequence at every point in − ∼ C. Thus Y ,Y has regular crossings in C. { x y} Remark 3.5. In both of the above examples, we checked the r.c. crossing condition only affine- locally, even though the definition is in terms of stalks. This is sufficient because if x1,...,xn is a regular sequence in a Noetherian ring R, then it remains a regular sequence in Rp for any prime ideal p R containing x ,...,x . ⊂ { 1 n} Definition 3.6. A subscheme Z X is called monomial if it is cut out by effective divisors which ⊂ are supported on a fixed s.n.c. divisor. By analogy, if Y ,...,Y has regular crossings, we will say { 1 n} Z is an r.c. monomial subscheme (with respect to Y ,...,Y ) if Z is cut out by divisors of the { 1 n} form a Y with a 0. As well, if β : X X is the blowup of X at an r.c. monomial subscheme, i i i ≥ → we willP call β (or just X) an r.c. monomial blowup (with respect to Y1,...,Yn ). e { }

Theorem 3.7 (Main Theorem)e . Let Y1,...,Yn have regular crossings on X. If D1,...,Dh are { } given by Dj = aijYi, where aij > 0, then there exists a sequence of r.c. monomial blowups at codimension 2 centersP

βn βn−1 β2 β1 X = Xn Xn 1 X1 X −−−−→ − −−−−→· · · −−−−→ −−−−→ such that (I + + I ) is r.c. monomial for each i and (I + + I ) e is locally D1 ··· e Dh OXi D1 ··· Dh OX principal.

Goward’s theorem [45, Theorem 2] is the analogous statement to Theorem 3.7 for monomial subschemes. The algorithm here is a direct generalization of Goward’s, and our method of proof follows his. We ﬁrst need to know that r.c. monomial blowups preserve regular crossings and so we verify this in section 3.3. Following this, we give the proof of Theorem 3.7 in section 3.4.

3.3 R.c. monomial blowups preserve regular crossings

Assume Y ,...,Y has regular crossings and n 2. Let D = a Y and D = b Y with { 1 n} ≥ 1 i i 2 i i ai,bj 0. If β : X X is the blowup of X at Y1 Y2, let E denoteP the exceptional divisorP in X ≥ → ∩ and let Y denote the proper transform of Y in X. We will verify that β D , which is supported i e i ∗ i e on E, Y ,..., Y , has regular crossings. { e1 n} e The proof will be by induction on the number of divisors. The base case is handled in Proposition e e 3.8 and the inductive step is handled in Proposition 3.10.

25 Proposition 3.8. Suppose Y ,Y has regular crossings on X. Let β : X X denote the blowup { 1 2} → of X at Y Y . Then E, Y , Y has regular crossings in X. 1 ∩ 2 { 1 2} e Proof. There are eight subsetse e of Y , Y ,E . The intersectionse corresponding to , Y , Y , { 1 2 } ∅ { 1 2} Y , Y ,E are empty, and those corresponding to Y , Y , E are effective Cartier. This { 1 2 } f f { 1} { 2} { } f f leaves the intersections corresponding to Y ,E and Y ,E . We show the result for Y ,E . f f { 1 } {f2 }f { 1 } It suffices to prove the result affine-locally. Since Y , Y are effective Cartier divisors on X, there f 1 f2 f is an affine open cover U = Spec R of X such that y is a local equation for Y on U and y { α α} 1,α 1 α 2,α is a local equation for Y . Now let U = Spec R be a member of such a cover and let (y ), (y ) R 2 1 2 ⊂ be principal ideals defining Y and Y , respectively. Then the blowup U of U centered at Y Y is 1 2 1 ∩ 2 Proj R[a ,a ]/(a y a y ), see [42, B.6.10]. Consider the open set D(a ) := [p] Spec R a / p 1 2 2 1 − 1 2 e2 { ∈ | 2 ∈ } which we can write as D(a ) = Spec R[a ]/(y a y ). 2 1 1 − 1 2

The pullback β∗Y1 is cut out by a1y2 in D(a2) and the exceptional divisor is cut out by y2, so the proper transform of Y is cut out in D(a ) by a . Since (R[a ]/(y a y ))/(a ) = R/(y ), 1 2 1 1 1 − 1 2 1 ∼ 1 and y2 is not a zerodivisor in R/(y1) by assumption, we have that a1,y2 is a regular sequence in R[a ]/(y a y ) corresponding to the intersection Y Y . The argument for D(a ) is analogous. 1 1 − 1 2 1 ∩ 2 1 The proof is completed by noting again (Remark 3.5) that localization preserves regular sequences, so that a ,y is a regular sequence in e for each p Y E. 1 2 OX,p ∈ 1 ∩ We note that as a scheme of ﬁnite type over a Noetherian ring,e X is Noetherian. In particular is a Noetherian local ring for all p X. In the proof of Proposition 3.10, we will need the OX,p ∈ following result.

Lemma 3.9. If R is a Noetherian local ring and x1,...,xr is a regular sequence of elements in the maximal ideal of R, then any permutation of x1,...,xr is again a regular sequence.

Proof. See [37, Theorem 17.2].

We will make repeated use of the lemma, along with the idea that if x1,...,xn is a regular sequence, then x ,...,x is a regular sequence for 1 k n. 1 k ≤ ≤ Proposition 3.10. Suppose Y ,...,Y has regular crossings on X. Let β : X X denote the { 1 n} → blowup of X at Y Y . Then E, Y ,..., Y has regular crossings in X. 1 ∩ 2 { 1 n} e e e e 26 Proof. By induction, assume E, Y1,..., Yn 1 has regular crossings. We will show that if A con- { − } tains any (possibly empty) subset of E, Y , Y , then the corresponding intersection is cut out by e { e1 2} a regular sequence at each point in an affine chart on the blowup. e e Let A E, Y1,..., Yn . We want Z AZ = so Y1, Y2 cannot be a subset of A. Assume ⊂ { } ∩ ∈ 6 ∅ { } then without loss of generality that Y / A. Assume also that Y A. Each element of Y ,...,Y e e 2 ∈ e e n ∈ { 1 n} is an effective Cartier divisor so again we can find a cover of X by open affines U = Spec R e e { α α} such that for each i, and each Y Y ,...,Y , we have a non-zerodivisor y R which cuts ∈ { 1 n} ∈ α out Y in U . Choose such a U = Spec R and let y R be a local equation for Y . We can then α i ∈ i write the blowup of this chart U = Proj R[a ,a ]/(a y a y ). As before, we work affine-locally 1 2 2 1 − 1 2 in D(a ) = Spec R[a ]/(y a y ), and we have that E is cut out by y and Y is cut out by a . 2 1 1 − 1 e2 2 1 1 Assume that A contains both E and Y . By induction we can find a regular sequence for 1 e A Y in the elements y ,a ,s ,...,s (where #(A Y )= k + 2) and so Lemma 3.9 ensures \{ n} { 2 1 1 k} e \{ n} that a ,y ,s ,...,s is a regular sequence at each e 1 2 1 k e p e Y . Let r be a local equation for Y on D(a ) such that we have a regular sequence in Y A Yn n 2 ∈∩ ∈ \{ } the elements r, y1,y2,s1,...,sk at each point p Y AY . Then by Lemma 3.9 these elements form e∈∩ ∈ a regular sequence in any order. Thus y2,r,s1,...,sn form a regular sequence in R/(t1), showing that a1,y2,r,s1,...,sn is a regular sequence at each point p Y AY . ∈∩ ∈ If A contains E but does not contain Y1, then we can use Lemma 3.9 again to get the regular sequence s ,...,s ,r,y ,a which shows that s ,...,s ,r,y is a regular sequence. Similarly, if A 1 n 2 1 e 1 n 2 contains Y1 but not E, we can rearrange to get s1,...,sn,r,a1,y2, which shows s1,...,sn,r,a1 is a regular sequence. Truncating this sequence again shows that s ,...,s ,r is a regular sequence. e 1 n This is the case where A contains neither E nor Y1.

In this chapter, we will only need the result ase stated in Proposition 3.10, but since the fact is true in greater generality, we provide Proposition 3.11 for completeness. In the following proof we use Proposition 3.10 as the base case and induct on the number of divisors that cut out the center for the blowup.

Proposition 3.11. Suppose Y ,...,Y has regular crossings on X. Let β : X X denote the { 1 n} → blowup of X at r Y , where r n. Then E, Y ,..., Y has regular crossings in X. ∩i=1 i ≤ { 1 n} e e e e

27 Proof. Let A E, Y1,..., Yn with Y1 A. We want Z AZ = so we require A Y2,..., Yr = ⊂ { } ∈ ∩ ∈ 6 ∅ ∩{ } . Each element of Y ,...,Y is an effective Cartier divisor so again we can find a cover of ∅ e{ 1 e n} e e e X by open affines U = Spec R such that for each α, and each Y Y ,...,Y , we have a { α α} ∈ { 1 n} non-zerodivisor y R which cuts out Y in U . Choose such a U = Spec R and let y R be a ∈ α α i ∈ local equation for Yi. We can then write the blowup of this chart

U = Proj R[a ,...,a ]/( a y a y i = j ) 1 r { j i − i j | 6 }

(again see [42, B.6.10]). Ase before, we work aﬃne-locally in

D(a ) = Spec R[a ,...,a ]/( y a y i = 1 ) 1 2 r { i − i 1 | 6 } where we have that E is cut out by y1, and Yi is cut out by ai for each i> 2. We can use induction on r with the base case Proposition 3.10. Then assume E, Y ,..., Y e { 1 n} r 1 has regular crossings in the blowup β : X X centered at − Y . If we let U denote the blowup ′ ′ → ∩i=1 i ′ e e of the chart U by β′ then

U ′ = Proj R[a1,...,ar 1]/( ajyi aiyj i = j ) − { − | 6 } with corresponding open chart

D′(a1) = Spec R[a1,...,ar 1]/( yi aiy1 i = 2,...,r 1 ). − { − | − }

Now we observe that D(a1) is related to D′(a1) by

D(a ) V(a ) = Spec(R[a ,...,a ]/( y a y i = 1 ))/(a ) 1 ∩ r 2 r { i − i 1 | 6 } r

= Spec R[a2,...,ar 1]/( yi aiy1 i = 1,r yr ) − { − | 6 } ∪ { }

= D′(a ) V(y ). 1 ∩ r

Recall that Y / A. The inductive hypothesis says that we can ﬁnd a regular sequence s ,...,s r ∈ 1 k corresponding to A, where k = #A. Then as usual we can localize at a point and rearrange to e get that y ,s ,...,s is a regular sequence at each point p D (a ). Then s ,...,s is a regular r 1 k ∈ ′ 1 1 k sequence at each point in D′(a1)/(yr)= D(a1)/(ar), so ar,s1,...,sk is a regular sequence at each point of D(a1).

28 3.4 Proof of main theorem

In [45], Goward deﬁnes an invariant (σ, τ) on the divisors in question. We adopt those deﬁnitions for the new context here:

Deﬁnition 3.12. Let D = n a Y and D = n b Y where Y ,...,Y has regular crossings 1 i=1 i i 2 i=1 i i { 1 n} in X and ai,bj 0. We deﬁneP P ≥

σ (D ,D ) = max ( a b , a b ), ( a b , a b ) ij 1 2 { | i − i| | j − j| | j − j| | i − i| } if a b and a b have opposite signs, and otherwise i − i j − j

σ (D ,D )=( , ), ij 1 2 −∞ −∞ where the max is taken lexicographically. Now we can deﬁne

σ(D ,D ) = max σ (D ,D ) Y Y = 0, i = j 1 2 { ij 1 2 | i ∩ j 6 6 } τ(D ,D )=# (i,j) σ (D ,D )= σ(D ,D ), i j 1 2 { | ij 1 2 1 2 ≤ } so σ(D1,D2) takes the value of the worst intersection in the support, and τ(D1,D2) counts how many intersections share this value.

These invariants are calculated for divisors on X and, after blowing up, for their pullbacks. We show that these calculations go the same way as in the simple normal crossings context and then outline the steps of the proof.

Proposition 3.13. Let D ,D be as deﬁned above. Then I + I is principal at p X if and 1 2 D1 D2 ∈ only if σ (D ,D )=( , ) whenever p Y Y . ij 1 2 −∞ −∞ ∈ i ∩ j Proof. Let x ,...,x be a regular sequence at p X corresponding to Y ,...,Y . Then let f = 1 n ∈ 1 n 1 u xa1 ...xan and f = u xb1 ...xbn be local equations for D and D , where u are units. 1 1 n 2 2 1 n 1 2 i ∈ OX,p I I Suppose D1 + D2 is not principal at p. Then (f1,f2) is not principal, so we have some i,j such that a b and a b have opposite signs. Thus σ (D ,D ) > 0. On the other hand, suppose i − i j − j ij 1 2 p Y Y and σ (D ,D ) > 0. Then a b and a b have opposite signs, so (f ,f ) is not ∈ i ∩ j ij 1 2 i − i j − j 1 2 I I principal and thus D1 + D2 is not principal at p.

29 I I I I I As a result of this proof, we see that if D1 + D2 is principal at p, then ( D1 + D2 )p =( Di )p for some i 1, 2 . So by induction, we have that if I + + I is principal at p, then ∈ { } D1 ··· Dh (I + + I ) =(I ) for some i 1,...,h . D1 ··· Dh p Di p ∈ { } We have left to show that blowing up at the chosen codimension 2 centers strictly reduces the invariant (σ, τ) and then that such blowups can be taken successively until (σ, τ)=( , ). −∞ −∞

Proposition 3.14. Let D1,D2 be as deﬁned above. Suppose we have (i,j) such that σij(D1,D2)= σ(D ,D ) > ( , ) and let β : X X be the blowup of X centered at Y Y . Then 1 2 −∞ −∞ → i ∩ j e (σ(D1,D2),τ(D1,D2)) > (σ(β∗D1,β∗D2),τ(β∗D1,β∗D2).

Sketch of proof. Assume that (i,j) = (1, 2) so that the blowup is centered at Y Y . The proof 1 ∩ 2 relies on calculations of σij(β∗D1,β∗D2), which depend only on the coeﬃcients ai,bi,aj,bj. The details can be found in the proof of [45, Thm 1]. Here we verify only that the calculations of

σij(β∗D1,β∗D2) from the s.n.c. case still go through with regular crossings. Let E X denote the exceptional divisor. Then ⊂

e Yi + E if i = 1, 2 β∗Yi = (Yi if i> 2 e as seen from the work done in Proposition 3.8.e Thus

β∗D1 =(a1 + a2)E + aiYi Xi e and

β∗D2 =(b1 + b2)E + biYi. Xi e Theorem 3.7 (Main Theorem). Let Y ,...,Y have regular crossings on X. If D ,...,D are { 1 n} 1 h given by Dj = aijYi, where aij > 0, then there exists a sequence of r.c. monomial blowups at codimension 2 centersP

βn βn−1 β2 β1 X = Xn Xn 1 X1 X −−−−→ − −−−−→· · · −−−−→ −−−−→ such that (I + +e I ) is r.c. monomial for each i and (I + + I ) e is locally D1 ··· Dh OXi D1 ··· Dh OX principal.

30 Proof. By the previous remarks, we can assume h = 2. Let Y1,...,Yn be the divisors in the support of D + D . If σ(D ,D )=( , ) we are done, so assume not. Then let 1 2 1 2 −∞ −∞

(i,j) = max (k,l) σ (D ,D )= σ(D ,D ) { | kl 1 2 1 2 } where the max is taken lexicographically. If we take the blowup β : X X centered at Y Y , 1 1 → i ∩ j then Proposition 3.14 gives that

(σ(D1,D2),τ(D1,D2)) > (σ(β∗D1,β∗D2),τ(β∗D1,β∗D2).

Now we note that β D , β D are divisors supported on E, Y ,..., Y and have regular crossings 1∗ 1 1∗ 2 { 1 n} by Proposition 3.10. Thus, (I + I ) e deﬁnes an r.c. monomial subscheme of X. D1 D2 OX e e We can repeat this process, with (σ, τ) decreasing at each iteration. Since (σ, τ) takes values in e N2 N, we must get (σ, τ)=( , ) after ﬁnitely many steps, yielding the desired sequence of × −∞ −∞ blowups.

31 CHAPTER 4

CLASSIFICATION OF THE MONOMIAL CREMONA TRANSFORMATIONS OF THE PLANE

4.1 Introduction

Let (x : y : z) be coordinates on P2. Let ϕ : P2 99K P2 be a monomial map

ϕ :(x : y : z) (xa11 ya12 za13 : xa21 ya22 za23 : xa31 ya32 za33 ). 7→

We can carry the information of this map in its exponent matrix Mϕ =(aij). We will be interested in birational monomial maps on P2. For such maps, the total degree of each monomial is constant, i.e., there is a δ such that the row sum is δ for each row of Mϕ. If the monomials xai1 yai2 zai3 share no common factors, we say that ϕ is written in reduced form. Throughout the chapter, we’ll assume that any monomial map P2 99K P2 comes in reduced form.

Of course, if ϕ is in reduced form, then Mϕ must have a 0 in each column. Thus, if Mϕ is the exponent matrix of a monomial Cremona transformation on P2, then up to swapping of rows and columns it has one of the forms 0 0 0 · · · 0 or 0 . · ·  · · 0 · · · · ·     A rational map on the projective plane has associated to it a tuple of numbers (γ0,γ1,γ2) called the multidegree (see section 4.3). A rational map is a Cremona map if and only if its multidegree is (1,d, 1), and if the map is monomial then d = δ as above. Theorem 4.1 below gives the complete list of monomial Cremona transformations of the plane. Note that (II) could actually have been included in (III) if we just allowed c = 0. We list it separately to emphasize that it is special in the sense that the exponent matrix has more than 3 zeroes.

Theorem 4.1. Let ϕ : P2 P2 be a monomial rational map with exponent matrix M . Then ϕ is → ϕ a Cremona transformation (with multidegree (1, δ, 1)) if and only if

32 (I) ϕ is the standard involution ϕ(x : y : z) (xy : xz : yz), or 7→

(II) Mϕ is of the form 0 0 δ M = 1 δ 1 0 , ϕ  −  0 1 δ 1 − or  

(III) Mϕ is of the form 0 0 δ M = a b 0 ϕ   c d e and the following equations are satisﬁed:  

(i) a + b = δ, c + d + e = δ, (ii) ad bc = 1, − (iii) b d> 0, a c> 0. ≥ ≥

Remark 4.2. Throughout the chapter we will allow ourselves to swap two rows or columns of Mϕ whenever convenient, as we’ve done here.

By combining theorem 4.1 with theorem 4.11, we get the following enumeration of monomial Cremona transformations of the plane.

Theorem 4.3. Let δ > 2. The monomial Cremona transformations ϕ : P2 99K P2 of degree δ are in one-to-one correspondence with the elements of (Z/δZ)×.

4.2 Some convex geometry

3 The rows of Mϕ give coordinates for points in R .

Deﬁnition 4.4. Let S = s ,...,s be a set of points in Rn. We denote the convex hull of S by { 1 k}

ch(S) := λ s + + λ s λ 0,λ + + λ = 1 { 1 1 ··· k k | i ≥ 1 ··· k } and the conical hull of S by

cone(S) := λ s + + λ s λ 0 . { 1 1 ··· n k | i ≥ }

33 In general, if R is an arbitrary subset of Rn, we deﬁne its convex hull to be

ch(R) := ch( a,b ) { } a,b R [∈ Here we are concerned with convex polyhedra, subsets P Rn which can be written ⊂

P = ch(S) + cone(S′) := a + b a ch(S), b cone(S′) . { | ∈ ∈ } For a polyhedron of dimension d, the faces of dimension 0,1 and d 1 are called vertices, edges and − facets, respectively.

Let := e ,e ,e be the standard basis in R3. If B { 1 2 3} A M = B ϕ   C   is the exponent matrix of a monomial rational map ϕ, then the Newton polyhedron of ϕ is N = ch( A, B, C ) + cone( ). This is a 3-dimensional, unbounded polyhedron with exactly one ﬁnite { } B facet: ch( A, B, C ). { } We will adopt the following notation for a convex polyhedron. If S = v ,...,v and S = { 1 j} ′ w ,...,w then we write v ,...,v ; w ,...,w to mean ch(S) + cone(S ). We say that the { 1 k} h 1 j 1 ki ′ order of the polyhedron is the minimal number S + S . Thus, the Newton polyhedron N is | | | ′| A, B, C; e ,e ,e . h 1 2 3i Finally, given a vector v R3, we say a point x N is minimal for v if it minimizes x v on N. ∈ ∈ · In particular, it will be useful to consider which vertices of N are minimal for ei.

4.2.1 Triangulations

We will need to triangulate N. In this chapter, a triangulation of a 3-dimensional convex polyhedron N is a set of 3-dimensional subsets of N such that P P = N and such that the T ⊔ ∈T order of P is 4 for all P (this makes P “simplicial”). ∈T A useful algorithm for doing this comes from [27]. We choose a distinguished vertex p ∈ A, B, C and construct a triangulation starting with p; e ,e ,e , that is, the positive orthant { } T h 1 2 3i translated to p. Now let F denote the facets of N which do not include the vertex p. If f F is p ∈ p a facet with order 3, then include ch( p f) in . Otherwise, we should construct a triangulation { }∪ T of f using the analogous procedure and for each t , include ch( p f) in . Tf ∈Tf { }∪ T

34 z

B y C

Figure 4.1: Schematic drawing of a Newton polyhedron

Example 4.5. In ﬁg. 4.1, where A, B, C are given by the matrix

0 0 3 M = 0 3 0 , ϕ   2 1 0   we can begin a triangulation of N with A; e ,e ,e . We then consider the facets which T h 1 2 3i do not include A. The only such face is B,C; e ,e . Since this face has order 4 > 3, we must h 1 2i triangulate it. We choose a vertex, say B, and take the cone B; e ,e at B. The remaining facet h 1 2i (of B,C; e ,e ) is C; e . Thus, the two polyhedra B; e ,e and B,C; e form a triangulation h 1 2i h 2i h 1 2i h 2i of B,C; e ,e . Taking the pyramid of these polyhedra at A completes the triangulation of N: h 1 2i

= A; e ,e ,e , A, B; e ,e , A, B, C; e . T {h 1 2 3i h 1 2i h 2i}

4.3 Computing multidegrees via polyhedra

Let Γ(ϕ) be the closure of the graph of ϕ in P2 P2. Then the class [Γ(ϕ)] A(P2 P2) is × ∈ ×

2 2 [Γ(f)] = γ0h2 + γ1h1h2 + γ2h1 where h ,h are the pullbacks of the hyperplane class c ( (1)) from P2 via the projections to the 1 2 1 O ﬁrst and second factors, respectively.

2 2 Deﬁnition 4.6. The multidegree of the rational map ϕ : P 99K P is the tuple (γ0,γ1,γ2).

35 Γ(ϕ) P2 P2 ⊂ ×

ϕ P2 P2

Figure 4.2: Graph of ϕ

The coeﬃcients γi satisfy

(i) γ = γ (ϕ) := #(ϕ 1(p)) for a general point p P2, 0 0 − ∈ (ii) γ = γ (ϕ) is the degree of (the closure of) ϕ 1(H) P2 for a general hyperplane H P2, 1 1 − ⊂ ⊂ (iii) γ = γ (ϕ) is the degree of the ﬁeld extension K(ϕ(P2)) K(P2). 2 2 ⊂

If ϕ is a monomial Cremona transformation, then γ0 = 1 because ϕ is generically one-to-one, 2 and γ2 = 1 because ϕ is birational. The degree of the image under ϕ of a hyperplane in P will be #(H ϕ 1(H )) = δ, the total degree of the monomials deﬁning ϕ. The conclusion is that a · − ′ monomial rational map ϕ on P2 is a Cremona transformation if and only if the multidegree of ϕ is (1, δ, 1).

4.3.1 Multidegrees as volumes

Let be a triangulation of N, the Newton polyhedron of ϕ. If S; V , we deﬁne π to be T h i∈T V the projection R3 V where V denotes the subspace spanned by V . We then say (in the → ⊥ ⊥ B\ language of [6, Sec. 3.1]) that the volume Vol( S; V ) is the normalized volume of π ( S; V ). h i V h i Our method for computing multidegrees is [6, Thm. 1.4], which we restate here using our notation. Letting T (i) = S; V #(S) = i + 1 be the polyhedra S; V whose projection {h i∈T | } h i under πV are of dimension i, Aluﬃ proved the following.

Theorem 4.7 (Aluﬃ). γi = P (i) Vol(P ). ∈T P Via this theorem, we have the following strategy. Find a triangulation of the Newton polyhedron

N associated to an exponent matrix Mϕ. If the volumes in the triangulation give (γ0,γ1,γ2) = (1, δ, 1), then ϕ is a Cremona transformation, otherwise not.

36 4.4 Newton polyhedra of Cremona transformations

In this section, we use the technology outlined in the previous sections to analyze the types of monomial Cremona transformations that exist, resulting in the proof of theorem 4.1. The speciﬁc problem is: given a 3 3 integer matrix, when is it the (reduced) exponent matrix of a monomial × planar Cremona transformation?

4.4.1 Main case

A We begin with the case where the exponent matrix Mϕ = B has the form C 0 0 0 0 δ · 0 = a b 0 . · ·    c d e · · ·     Note, we assume that c,d are both non-zero, since otherwise we would be in the other case (cf. section 4.1).

The following fact about Mϕ will be very useful.

Lemma 4.8. det(M ) = δ. | ϕ | Proof. See [44, Prop. 3.1] or [32, Prop. 1].

If Mϕ is going to be an exponent matrix for a monomial Cremona transformation then we must have a + b = δ = c + d + e (4.1) and by lemma 4.8 we must also have δ(ad bc) = δ, so ad bc = 1. By swapping columns 1 | − | − ± and 2 if necessary, we may assume ad bc = 1 (4.2) − which uniquely determines the edges of the Newton polyhedron N associated to Mϕ (consider the projection of the points B,C to the xy-plane), an example of which is shown in ﬁg. 4.3. Begin constructing the triangulation with B; e ,e ,e . The facets of N which do not contain T h 1 2 3i B are A; e ,e , A; e ,e , A, C; e , h 1 3i h 2 3i h 1i

37 z

B y

Figure 4.3: Another Newton polyhedron so we have = B; e ,e ,e , A, B; e ,e , A, B; e ,e , A, B, C; e . T {h 1 2 3i h 1 3i h 2 3i h 1i} The polyhedron B; e ,e ,e projects to the point B, which has V ol(B)=1= γ . The projection h 1 2 3i 0 of A, B; e ,e onto the x-axis is the interval (0,a), which has Vol((0,a)) = a. The projection of h 2 3i A, B; e ,e onto the y-axis is the interval (0,b), which has Vol((0,b)) = b and as we would hope, h 1 3i we get γ1 = a + b = δ. Finally, we consider A, B, C; e which should be projected onto the yz-plane. The projection h 1i is a triangle with vertices (0,δ), (b, 0), (d,e), and its volume is 0 δ 1 1 Vol( A, B, C; e )= (2!)( ) b 0 1 = δd + be δb h 1 2! | − | d e 1

So since we should have γ = 1 we have the necessary condition 2

δd + be δb = 1. (4.3) | − | Notice though, that if we use a = δ b and c = δ d e in the determinant equation ad bc = 1, − − − − we get (δ b)d b(δ d e)= δd + be δb = 1. − − − − − This shows that eq. (4.3) is equivalent to the requirement that ad bc = 1 as long as we require − also that eq. (4.1) be satisﬁed, completing the proof of the following lemma.

38 Lemma 4.9. If c> 0,d> 0, then the exponent matrix

0 0 δ M = a b 0 ϕ   c d e   deﬁnes a monomial Cremona transformation with multidegree (1, δ, 1) if and only if the following equations are satisﬁed:

(i) a + b = δ, c + d + e = δ,

(ii) ad bc = 1. − For a quick payoﬀ, if we set a = δ 1, then we ﬁnd a uniquely determined exponent matrix: − 0 0 δ M = δ 1 1 0 ϕ  −  δ 2 1 1 −   and one can check that this yields a monomial Cremona transformation with inverse given by the exponent matrix 1 δ 1 0 − M −1 = 0 0 δ . ϕ   1 δ 2 1 −   4.4.2 Other cases

Now assume Mϕ has the form

0 0 a b · · 0 = c 0 d . · ·   0 e f 0 · ·     Here we have a,b,c,d,e,f Z 0 with ∈ ≥

a + b = c + d = e + f = δ. (4.4)

First note that lemma 4.8 in this case gives ade + bcf = δ. Rewriting using (4.4), we get | |

ade + bcf = a(δ c)e +(δ a)c(δ e)= δ(δc + ae ac ce) − − − − − so δc + ae ac ce = 1. (4.5) | − − |

39 No additional zeroes. Assume a,b,c,d,e,f are all nonzero. Then A is the unique minimal vertex for e1∗, B is the unique minimal vertex for e2, and C the unique minimal vertex for e3. This determines the ridge structure of the Newton polyhedron N. An example of this structure is shown in ﬁg. 4.4, where ϕ(x : y : z)=(yz : xz : xy).

C x

Figure 4.4: Newton polyhedron for the standard involution

If we begin our triangulation with the distinguished point A, we get

= A; e ,e ,e , A, B; e ,e , A, C; e ,e , A, B, C; e . T {h 1 2 3i h 1 3i h 2 3i h 1i}

The projection of A along all three directions ei is a point with volume γ0 = 1. The projection of ch(A, B) along e1,e3 is the interval (0,a), and the projection of ch(A, C) along e2,e3 is the interval

(0,e). The sum of these two volumes is γ1 = a + e. We want γ1 = δ, so this gives a new condition:

b = e and a = f. (4.6)

The projection of ch(A, B, C) onto the yz-plane has normalized volume

a b 1 det 0 d 1 = ad + bf df   | − | f 0 1

  and we want γ2 = 1 so we should require

ad + bf df = 1 (4.7) | − | ∗By ‘minimal vertex for v’, we mean x · v is minimized by the row vector x.

40 which becomes ab = 1 by (4.6). Then, a =1= b and so e =1= f by (4.6) and c =1= d by | | (4.5). Thus, 0 1 1 M = 1 0 1 . ϕ   1 1 0   f = 0. If f = 0, then e = δ, so we have ade + bcf = δad = δ which implies a = 1,d = 1. | | | | Then b = δ 1 and c = δ 1, so − − 0 1 δ 1 − M = δ 1 0 1 . ϕ  −  δ 0 0   The Newton polyhedron in this case is shown in ﬁg. 4.5.

B y

C x

Figure 4.5: Newton polyhedron when f = 0

Our previous triangulation is still a triangulation here. The facet B,C; e ,e can be trian- T h 1 3i gulated with B; e ,e , B,C; e {h 1 3i h 1i} which were the facets we used before, so the algorithm still generates

= A; e ,e ,e , A, B; e ,e , A, C; e ,e , A, B, C; e . T {h 1 2 3i h 1 3i h 2 3i h 1i}

Then to get a Cremona map, we should still require ad + bf df = 1, and indeed this becomes | − |

1 1 + 0 0 = 1. | · − |

41 This also takes care of the case where one of a,c,d is zero, since we can take the resulting matrix to one of the proper form by row and column swaps. For instance, if d = 0, we get 0 a b 0 a b 0 b a c 0 d = c 0 0 e 0 f       e f 0 e f 0 c 0 0       by swapping rows 2,3 and swapping columns 2,3.

e = 0. If e = 0, then f = δ, so we have ade + bcf = δbc = δ, so b = 1,c = 1. Then a = δ 1 | | | | − and d = δ 1, so − 0 δ 1 1 − M = 1 0 δ 1 . ϕ  −  0 δ 0 But this is the same as in the previous case, just swap rows 1,2 and columns 1,2. So this section concludes the analysis of exponent matrices which are not of the form of lemma 4.9.

Lemma 4.10. Let ϕ : P2 P2 be a monomial rational map where M is not in the form of → ϕ lemma 4.9. Then ϕ is a Cremona transformation (with multidegree (1, δ, 1)) if and only if 0 δ 1 1 0 1 1 − M = 1 0 δ 1 or M = 1 0 1 ϕ  −  ϕ   0 δ 0 1 1 0     (up to row and column swaps).

4.5 Number of monomial Cremona transformations

Theorem 4.11. For δ 3, there are exactly φ(δ) exponent matrices, as described in theorem 4.1, ≥ where φ is Euler’s totient function.

Proof. There is exactly one exponent matrix of the form 0 0 δ 1 δ 1 0 ,  −  0 1 δ 1 −   so we must show that there are φ(δ) 1 exponent matrices of the form described by (III) in − theorem 4.1. These are 0 0 δ a b 0   c d e satisfying  

42 (i) a + b = δ, c + d + e = δ,

(ii) ad bc = 1. − Substituting d = δ c e and b = δ a into (ii), we get − − −

δa δc ae = 1 (4.8) − − which we rearrange to get δ(a c)= ae + 1. − This implies δ (ae + 1) which we can write as ae δ 1 mod δ. Then, for each a (Z/δZ) we ≡ − ∈ × can set e = a 1 (δ 1), utilizing the natural correspondence of (Z/δZ) 1,...,δ . However, we − − × ⊂ { } cannot allow e = δ 1 because this would require c + d = 1, which is impossible by (ii) and the − fact that c,d > 0. If a,e are determined, then δr = ae + 1 for some positive integer r, and we can let c = r a. − Thus, there is a unique solution a,c,e to (4.8) for each value of a Z/δZ 1 . Since this set has { } ∈ \{ } order φ(δ)-1, the proof is complete.

Example 4.12. For δ = 5, we have φ(5) = 4 and indeed we have the Cremona transformations given by 0 0 5 0 0 5 0 0 5 0 0 5 1 4 0 , 2 3 0 , 3 2 0 , 4 1 0 .         0 1 4 1 2 2 1 1 3 3 1 1 Example 4.13. In contrast to the previous example,  where δwas prime, consider δ = 12 which has φ(12) = 4 with totatives 1, 5, 7, 11 . Then the exponent matrices for the Cremona transformations { } are 0 0 12 0 0 12 0 0 12 0 0 12 1 11 0 , 5 7 0 , 7 5 0 , 11 1 0 .         0 1 11 2 3 7 4 3 5 10 1 1 Example 4.14. By theorem 4.1, there are exactly 2 exponent matrices having δ= 2. These are

0 1 1 0 0 2 1 0 1 , 1 1 0     1 1 0 0 1 1     and as always, this statement is up to row/column swapping.

43 4.5.1 Similarity

We could consider these maps up to permutation similarity, where two exponent matrices M,M ′ T yield the same map if and only if there exists a permutation matrix Z such that M ′ = Z MZ. In other words, M,M are equivalent if and only if there exists σ S , the symmetric group on ′ ∈ 3 1, 2, 3 , such that M = σ⋆M where S acts on M (Z) by permuting the rows and columns of { } ′ 3 3 M ′. For instance, if σ = (1 2), then σ⋆M ′ is the matrix obtained from M ′ by swapping the ﬁrst and second rows and then swapping the ﬁrst and second columns.

Since #(S3) = 6 and S3 acts freely on the subset of M3(Z) deﬁned by theorem 4.1, under this new equivalence relation we have exactly 6 non-similar matrices for each matrix described by the theorem. So in the new context, we say that there are exactly 12 monomial Cremona transformations on P2 having δ = 2. Under this equivalence relation we have the following.

Corollary 4.15. For δ 3 there are exactly 6 φ(δ) monomial Cremona maps on P2 up to ≥ · permutation similarity.

To motivate this perspective, consider the problem of determining γ = δ for ϕn = ϕ ϕ ϕ 1 ◦ ◦ · · · ◦ for some monomial Cremona transformation ϕ. Let ϕ1,ϕ2 be the two maps deﬁned by

0 0 5 3 1 1 M = 4 1 0 , M = 0 0 5 , 1   2   3 1 1 4 1 0     that is, M2 is given by acting with (123) on the rows of M1, Then, letting dn denote δ = γ1 corresponding to ϕn, we have d1 d2 d3 d4 d5 ϕ1 5 15 40 105 275 ϕ2 5 9 13 17 21 so from this dynamical perspective, these two maps should not be considered “the same”.

44 CHAPTER 5

THE CHERN-MATHER CLASS OF THE MULTIVIEW VARIETY

This chapter is joint work with Daniel Lowengrub.

5.1 Introduction

Suppose that a collection of cameras are used to generate images of a scene. The problem of triangulation is to deduce the world coordinates of an object from its position in each of the camera images. If we assume that the image points are given with infinite precision, then two cameras suffice to determine the world point. However, due to the many sources of noise in real images such as pixelization and distortion, there typically will not be an exact solution and we will instead try to find a world point whose picture is “as close as possible” to the image points. More precisely, suppose the cameras are C ,...,C and the image points are p ,...,p R2. 1 N 1 n ∈ The goal is to find a world point q R3 that minimizes the least squares error ∈ N error(q)= (C (q) p )2. i − i Xi=1 One application is the problem of reconstructing the 3D structure of a tourist attraction based on millions of online pictures. It is difficult to obtain the precise configuration of any single camera, so it would not make sense to use only a small subset of them and disregard the rest. A better approach is to solve an optimization problem which incorporates as many of the cameras as possible. This technique was used in [1] to reconstruct the entire city of Rome from two million online images. Since the camera function C : R3 R2 is not linear, the standard method for solving the i → triangulation problem is to first find the critical points of error(q) (e.g, with gradient descent), and then select the one with the smallest error. In order to gauge the difficulty of this problem, it is important to be able to predict the number of critical points that we expect to find for a given configuration of cameras.

45 The goal of this chapter is to give an explicit expression for the number of critical points of error(q) as a function of the number of cameras N. In fact, we compute this expression for a variation of the problem in which we allow the world points to take complex values, and we allow 3 3 these points to be in the projective space PC as opposed to the aﬃne space C . Our main result is that the number of critical points of error(q) is polynomial in the number of cameras.

3 Theorem 5.1. The number of critical points of error(q) on PC is equal to

p(N) = 6N 3 15N 2 + 11N 4 − − where N is the number of cameras.

Note that our reformulation of the problem only increases the number of possible critical points. One can solve the original problem by first finding these points, and then discarding the ones that are not in R3. For a similar reason, the polynomial p(N) is an upper bound on the number of critical points in the classical triangulation problem. In [72], a detailed investigation of the Lagrange multiplier equations which define the critical points is used to compute the number of such points for N 7. ≤ Based on these results, it was conjectured in [35, Conjecture 3.4] that the number of points should grow as the following polynomial:

9 21 q(N)= N 3 N 2 + 8N 4. 2 − 2 −

We note that our upper bound p(N) is fairly close. In order to compute the number p(N), we take a slightly diﬀerent perspective on the function error(q). By combining the cameras C : R3 R2 we obtain a rational map i →

φ : R3 R2N . →

After passing to the complex numbers and taking the projective closure we obtain a rational map φ : P3 P2N . C → C The image of this map is a three-dimensional variety MV P2N which is known as the N ⊂ multiview variety. We can now interpret the error function error(q) as measuring the distance

46 between a point q P2N and MV . With this formulation, the number of critical points is known ∈ N as the Euclidean distance degree of the variety MVN . The notion of ED degree was introduced in [35], and the authors remark in [35, ex 3.3] that the triangulation problem was their original motivation for this concept. In particular, by using results from [35] we prove in section 5.5 that this number can be computed in terms of the more familiar Chern-Mather class cMa(MVN ) which we write in this chapter as M c (MVN ) to make subscripts easier to read. One advantage of this approach is that it depends only on the geometric properties of MVN , and not on the speciﬁc features of the deﬁning equations.

Another advantage is that it reduces most of the difficulty to local calculations on MVN . One common way of calculating the Chern-Mather class of a singular variety X is to first find a resolution f X˜ X −→ and then analyze the singularities of f in order to compare the Chern class c(X˜) to the Chern- Mather class cM (X)

In our situation, it is natural to build a resolution of MVN by resolving the rational map φ. In section 5.3, we construct such a resolution

φ˜ : P˜3 MV → N and calculate its Chow ring and Chern class. ˜3 In order to compare the Chern class of P to the Chern-Mather class of MVN , we use the theory of higher discriminants which was introduced in [61]. One aspect of this theory is that it specifies which parts of the singular locus of X we need to understand in order to relate c(X˜) to cM (X). A precise statement is given in proposition 5.11. As we show in proposition 5.12, the higher discriminants of φ˜ are surprisingly nice. Specifically, M it turns out that in order to calculate c (MVN ), we only have to compute the Euler obstruction of a single point x MV . ∈ N Moreover, in section 5.5.2 we show that after intersecting MVN with a hyperplane at x, the resulting surface singularity (S,x) is taut. In particular, the Euler obstruction EuMVN (x) is determined by the resolution graph of x in S. This allows us to use the enumerative properties of P˜3 that are worked out in section 5.3 to compute EuMVN (x). In the final section, we put these pieces together and obtain the polynomial p(N).

47 5.2 Deﬁnitions and notation

Let P be a 3 4 matrix with values in R. We consider each row l as an aﬃne function on R3. × Explicitly, l sends a vector v = (x, y, z) to the dot product of l and (x,y,z, 1). We denote these functions by f, g and h. 3 2 The matrix P deﬁnes a rational map φP : R 99K R :

v (f(v)/h(v),g(v)/h(v)) 7→ which corresponds to the operation of mapping the “world coordinates” R3 to the “image coordinates” R2. In other words, it describes the process of taking a picture of the world with a camera whose parameters are encoded in P . It is not hard to prove that this description of a camera is equivalent to the pinhole camera model. In particular, the camera has a position called the camera center and is pointing in a certain direction. The plane defined by the camera center and direction is called the camera plane. It turns out that with the above notation, the camera plane is the plane defined by the ideal (h), and the camera center is the point defined by (f,g,h). For the purposes of this chapter, this observation will be taken as a definition.

Now, suppose that we have a collection of cameras P1,...,PN . By taking a picture of the world with each of the cameras, we obtain a rational map:

φ φ : R3 99K R2 R2 = R2N P1 ×···× PN ×···× ∼

This map clearly extends to the complex numbers, giving us a rational map from C3 99K C2N .

Furthermore, by clearing the denominators in the deﬁnition of the maps φPi we obtain a rational map 3 99K 2N φ : PC PC deﬁned by

φ ([x : y : z : w])=(f1h2 ...hN : g1h2 ...hN : : h1 ...hN 1gN : h1 ...hN ) . (5.1) ··· −

The scheme theoretic image of this map is called the multiview variety associated to the cameras

P1,...,PN .

48 Example 5.2. Consider the following three cameras:

1 0 0 1 0 1 0 1 1 0 0 1 P = 0 0 1 1 ,P = 0 0 1 1 ,P = 0 1 0 1 . 1   2   3   0 1 0 0 1 0 0 0 0 0 1 0       The associated rational map is

φ([x : y : z : w]) = [(x w)xz :(z w)xz :(y w)yz :(z w)yz :(x w)xy :(y w)xy : xyz]). − − − − − −

We say that a collection of cameras is in general position if the hyperplanes deﬁned by the linear functions f ,g ,h ,...,f ,g ,h associated to the rows of the camera matrices are in general { 1 1 1 N N N } position.

L12

H1 q2 q1 p123

L23

q3 L13 H3

Figure 5.1: Schematic of three cameras

Finally, we will use the following notation throughout the chapter (see ﬁgure 5.1). The camera plane of the i-th camera will be denoted by Hi and the center of the i-th camera will be denoted by q . Also, we deﬁne L = H H for all 1 i

5.3 A resolution of the multiview variety

In this section we describe a resolution of the multiview variety associated to N cameras in general position. It is obtained as an iterated blow up along smooth centers. We then apply

49 standard theorems to compute a presentation of the Chow ring of the resolution, and identify a couple of important ring elements.

Let P1,...,PN be camera matrices for a collection of N cameras in general position, and let

φ : P3 99K P2N be the corresponding rational map. We denote the associated multiview variety by MV P2N . N ⊂ Proposition 5.3. The base locus B of φ is the reduced scheme supported on the union of the camera centers q ,...,q and the lines L = H H for all 1 i

We now analyze the scheme structure of B in a neighborhood of the point p123 =(x, y, z). First of all, recall that the i-th camera contributes the two equations fi j=i hj and gi j=i hj to the · 6 · 6 ideal of B. Q Q By our genericity assumptions, all of the f ’s, all of the g ’s, and h for i 4 are invertible in i i i ≥ some Zariski neighborhood of p123. This implies that in a neighborhood of p123, the ideal of B has the form: (xy, xz, yz).

Thus, the ideal deﬁned by this scheme is reduced and supported on the coordinate axes. The same argument shows that all of the lines Lij in the base locus have the reduced scheme structure. A similar argument implies the points qi are reduced.

5.3.1 Constructing a resolution of φ

3 In this section we construct a resolution of MVN in two stages. First, we blow up P at the points q ,...,q and at the points p for all 1 i

b : Y P3. 1 1 →

Let L˜ Y denote the proper transform of L . Note that these proper transforms are disjoint ij ⊂ 1 ij lines in Y1.

50 For the second step, we blow up each of the lines L˜ij and obtain a resolution

b : Y Y 2 2 → 1

Let us denote Y by P˜3, and denote the composition b b by π. Since the pullback of the base 2 1 ◦ 2 ψ locus π 1(B) is a Cartier divisor on P˜3, there exists a canonical map P˜3 Bl P3 which ﬁts into − −→ B the following diagram: 3 ψ / 3 P˜ BlBP π BlB φ b " # P3 / P2N φ were b is the blowup map and BlBφ is the resolution of the rational map φ. Finally, we deﬁne φ˜ = Bl φ ψ. Since P˜3 is smooth, we thus obtain the following resolution of B ◦ MVN : P˜3 φ˜ π % 3 / 2N P MVN P φ ⊂ By an abuse of notation, we will sometimes think of φ˜ as a map to P2N , and other times as a map to MVN .

5.3.2 The Chow ring of P˜3

Since P˜3 is an iterated blowup of P3 along smooth centers, we can use standard theorems to compute its Chow ring. We will use a statement in [54] which we state here for convenience.

Theorem 5.4. [54, Appendix, Thm. 1] Let X i Y be a closed embedding of smooth schemes. −→ Let Y˜ be the blowup of Y along X and let X˜ denote the exceptional divisor. Suppose the map i :A (Y ) A (X) is surjective. Then, A (Y˜ ) is isomorphic to ∗ • → • •

A•(Y )[T ] (P (T ),T ker(i )) · ∗ where P (T ) A (Y )[T ] is a degree d polynomial whose constant term is [X], and whose re- X/Y ∈ • striction to X is the Chern polynomial of NX/Y . In other words,

d d 1 i∗PX/Y (T )= T + c1(NX/Y )T − + + cd 1(NX/Y )T + cd(NX/Y ). ··· −

51 The isomorphism is induced by the map f :A (Y ) A (Y˜ ), and by sending T to the class of ∗ • → • − the exceptional divisor.

The polynomial PX/Y is called the Poincarépolynomial of X in Y . b1 b2 By applying theorem 5.4 first to Y P3 and then to P˜3 = Y Y we find that A (P˜3) is a 1 −→ 2 −→ 1 • quotient of the polynomial algebra

A = Z[ h Qi 1 i N Pijk 1 i

[˜q ]= Q , [˜p ]= P , [L˜ ]= T . i − i ijk − ijk ij − ij In the next section, we will need to evaluate the degree map

deg : A3(P˜3) Z. → Since P˜3 is irreducible, A3(P˜3) has rank one. In addition, deg(h3) = 1. This means that calculating the degree map is equivalent to expressing every monomial α A3(P˜3) as a multiple of h3: ∈ α = deg(α) h3. · To simplify the calculation, note that product of two generators that correspond to disjoint subschemes of P˜3 is zero. For example, Q P = 0 for all i, j, k and l. i · jkl 3 Thus, the main difficulty is dealing with self intersections such as Tij. In order to deal with these, we will calculate the Poincarépolynomials of q Y , p Y and L Y . By theorem i ⊂ 1 ijk ⊂ 1 ij ⊂ 2 5.4, this will give us relations involving the self intersections, which in this case turn out to suffice for the degree calculation. Since q Y is a point, its Poincarépolynomial is i ⊂ 1 3 3 Pqi/Y1 (Qi)= Qi + h , and similarly, 3 3 Ppijk/Y1 (Pijk)= Pijk + h .

52 Finally, note that L Y is a line that passes through N 2 blown up points. We deduce ij ⊂ 1 − from this that P (T )= T 2 2(N 3)hT + h2 + P 2 . Lij /Y2 ij ij − − ij ijk k/ i,j ∈{X } 5.3.3 The Chern class of the resolution

3 3 2N In this section we compute c(P˜ ) as an element of A•(P˜ ) and ﬁnd its pushforward to P (proposition 5.8). Our main tool will be the following proposition.

Proposition 5.5. [42, Example 15.4.2] Let Y be a smooth scheme and X Y be a closed smooth ⊂ subscheme with codimension d. Consider the following blowup diagram.

j X˜ / Y˜

g f X / Y i

Suppose that c (N )= i c for some c Ak(Y ), and that c(X)= i α for some α A (Y ). Let k X/Y ∗ k k ∈ ∗ ∈ • η = c ( (X˜)). Then, 1 OY˜ c(Y˜ ) f ∗c(Y )= f ∗(α) β − · where d d i β = (1+ η) (1 η) f ∗cd i f ∗cd i. − − − − Xi=0 Xi=0 One takeaway of this proposition is that the Chern class of the blowup along a disjoint union of subvarieties is obtained by summing over contributions from the individual components.

Proposition 5.6. The Chern class of the resolution P˜3 is equal to

3 4 c(P˜ )=(1+ h) + αi + βij + γijk 1 i N 1 i

α = (1 Q )(1 + Q )3 1, i − i i − β = (1+ h)2 [(1 T )((1 + T )( 2(N 3)h)+(1+ T )2) (1 2(N 3)h)], ij · − ij ij − − ij − − − γ = (1 P )(1 + P )3 1. ijk − ijk ijk −

53 Proof. Our strategy will be to use proposition 5.5 to compute the contributions to the Chern class of each of the varieties that are blown up during the construction of P˜3. 3 We ﬁrst apply proposition 5.5 to the situation where Y = P and X = qi for some i. In this case, we can take c0 = 1, ck = 0 for k > 0 and α = 1. By proposition 5.5 the blowup at qi will contribute α = (1 Q )(1 + Q )3 1. i − i i −

Similarly, γijk represents the contribution from the blowup of the point pijk. Finally, we compute the contribution from the blowup along a line f : L ֒ Y . Since L ij → 1 ij passes through N 2 of the blown up points in Y , a quick calculation shows that we can take − 1 c = 1, c = 2(N 3)h, and the rest to be zero. In addition, since L = P1, we can take 0 1 − − ij ∼ 2 α = (1+ h) . This implies that the contribution coming from Lij is

β = (1+ h)2 [(1 T )((1 η)( 2(N 3)h)+(1+ T )2) (1 2(N 3)h)]. ij · − ij − − − ij − − −

3 We now compute the pullback of c ( 2n (1)) in A (P˜ ) along the map φ˜. 1 OP • 3 Lemma 5.7. The pullback of c ( 2n (1)) to P˜ is 1 OP 3 φ˜∗(c ( 2n (1))) [P˜ ]= N h + 2 P + Q + T . 1 OP ∩ · · ijk i ij 1 i

f ∗ (1) = π∗(L) ( E) O ⊗O − where E X˜ is the exceptional divisor. ⊂ In our case, one can show by a local calculation that the preimage of the base locus of the camera map to P˜3 has class

c ( ( E)) [P˜3] = 2 P + Q + T , 1 O − ∩ · ijk i ij 1 i

54 so that c (φ˜ ( (1))) = c (π (N)) + c ( ( E)) gives the stated expression. 1 ∗ O 1 ∗O 1 O − We can now compute the pushforward φ˜ c(P˜3) as an element of the Chow ring of P2N . ∗

Proposition 5.8. The pushforward to P2N of c(P˜3) is

N N N N φ˜ c(P˜3)= N 3 (4 + N) N 2 [P3]+ 4N 2 2 6 2N [P2] ∗ − 2 − − 3 − 3 − 2 − N N N + 6N +(N 4) [P1]+ 4 + 2N + 2 + 2 [P0] − 2 3 2 3 3 3 Proof. Since we have already calculated c(P˜ ) A (P˜ ) and φ˜ (c ( 2n (1))) A (P˜ ), the calcu- ∈ • ∗ 1 OP ∈ • lation of φ˜ c(P˜3) is reduced to calculating the degrees of the intersections ∗

k 3 π∗(c ( 2n (1))) c(P˜ ) 1 OP ∩ for 0 k 3. Using the relations in A (P˜3) that we described in section 5.3.2, the result follows ≤ ≤ • by a direct calculation.

5.4 Higher discriminants

Higher discriminants, introduced in [61], provide a framework in which to study the singularities of a map. In particular, we will use them to understand how the Chern class of P˜3 computed above pushes forward along φ˜. We now recall the deﬁnitions from [61], and phrase them in a way that will be easiest to use in our context.

Deﬁnition 5.9. Let f : Y X be a map of smooth manifolds. The i-th higher discriminant of → the map f is the locus of points x X such that for every i 1 dimensional subspace V T X, ∈ − ⊂ x there exists a point y f 1(x) such that: ∈ −

V,f TyY = TxX h ∗ i 6

We denote the i-th higher discriminant by ∆i(f).

For example, a point x X is in ∆1(f) if and only if it is a critical value of f. Indeed, according ∈ to the deﬁnition this happens exactly when there is a point y f 1(x) whose Jacobian ∈ −

J(f) : T Y T X y y → x

55 is not surjective. On the other extreme, x ∆dim(X)(f) if and only if for every codimension one subspace ∈ V T X, there exists a point y f 1(x) that satisﬁes: ⊂ x ∈ −

f TyY V. ∗ ⊂

It is instructive to consider the blow down map: f : Y = Bl P2 P2. For every point p → 1 1 y Ep = f − (p), f TyY is one dimensional. This means that p ∆ (f). In addition, it is not ∈ ∗ ∈ hard to see that for every one dimensional subspace V T P2, there is a point y E such that ⊂ p ∈ p 2 f TyY = V . This implies that p ∆ (f). ∗ ∈ Lemma 5.10. [61, Rem. 3] Let Y X be a proper map of smooth schemes. Then all of the → higher discriminants of f are closed, and we have the following stratiﬁcation of X:

∆dim(X)(f) ∆2(f) ∆1(f) X. ⊂···⊂ ⊂ ⊂

Furthermore, codim(∆i(X)) i. ≥ The signiﬁcance of the higher discriminants is that they tell us which strata appear when writing f ✶Y in the basis of Euler obstruction functions on X. For background on Euler obstructions we ∗ recommend [60].

Proposition 5.11. [61, Cor. 3.3] Let f : Y X be a proper map of complex varieties. Let ∆i,α → { } be the codimension i components of ∆i(f). Then,

✶ i,α f Y = η Eu∆i,α ∗ X for some integers ηi,α.

5.4.1 Higher discriminants of the resolution φ˜

In this section we describe the higher discriminants of the map

φ˜ : P˜3 MV P2N . → N ⊂

Since the deﬁnition of higher discriminants assumes that the source and target are smooth, in this section we consider φ˜ as a map to P2N .

56 Let X = P1 MV denote the image of the proper transform of the camera plane of the i-th i ∼ ⊂ N camera. The restriction of φ˜ to the complement of the preimage of the Xi’s is an isomorphism, which means that the set theoretic singular locus of φ˜ is contained in the disjoint union X . ∐i i The following proposition describes the higher discriminants of φ˜.

Proposition 5.12. The higher discriminants of φ˜ are given as follows:

∆2N 3(φ˜)=∆2N 2(φ˜)= MV • − − N ∆2N 1(φ˜)= X • − ∐i i ∆2N (φ˜)= • ∅ To prove this proposition, we use the following lemma, which follows almost immediately from the deﬁnition of the higher discriminants.

Lemma 5.13. Let f : Y X be a map of smooth complex algebraic varieties. Let C X be a → ⊂ smooth curve. Suppose that the restriction of f to C has no critical values. Then C ∆dim(X) = . ∩ ∅ Proof. Since f has no critical values, for every point x C and every point y f 1(x) the one |C ∈ ∈ − dimensional space TxC TxX is contained in f TyY . Therefore, if V TxX is the orthogonal ⊂ ∗ ⊂ complement to TxC, then f TyY is not contained in V . By deﬁnition, this implies that x / ∗ ∈ ∆dim(X)(X).

We apply this lemma to each of the P1’s X P2N . Let f : Y P1 = X denote the restriction i ⊂ → ∼ i ˜ 2 N 1 of φ to Xi. Then Y is isomorphic to the blowup of P at 1 + 2− points: q = qi and pijk for j,k = i. 6 The map f is obtained as follows. First, let

g : Bl P2 P1 q → be the resolution of the projection away from q. Then, let

h : Bl (P2) Bl (P2) q,pijk → p be the blowup along all of the points p for j,k = i. ijk 6 Finally, we claim that f = g h. In particular, f has no critical values. According to the lemma, ∼ ◦ this proves proposition 5.12.

57 5.5 The Chern-Mather class of the multiview variety

In this section we compute the Chern-Mather class of MVN using the theory of higher discriminants. We then use the result to determine the ED degree of MVN .

5.5.1 The basic setup

By propositions 5.11 and 5.12, there exists and integer α such that

N ˜ ✶ φ ( ˜3 )=EuMVN + α EuXi . (5.2) ∗ P · Xi=1 At a general point x X , χ(φ˜ 1(x)) = χ(P1) = 2 and Eu (x) = 1. This implies that ∈ i − Xi

2=Eu (x)+ α α = 2 Eu (x). MVN ⇒ − MVN

For the moment, suppose we knew the Euler obstruction EuMVN (x). Then, by taking the Chern- 1 Schwartz-MacPherson class (see [60]) of both sides of equation 5.2 and recalling that Xi ∼= P we obtain ˜ ˜3 M M 1 φ (c(P )) = c (MVN ) + (2 EuMVN (x))c (P ). (5.3) ∗ − Since we have already calculated φ˜ (c(P˜3)) for all N, this would give us the Chern-Mather class of ∗ the multiview variety MVN .

5.5.2 Calculating EuMVN (x)

To compute EuMVN (x), ﬁrst note that we can intersect MVN with a general hypersurface H passing through x. As a result, we obtain a surface singularity:

x S = MV H. ∈ N ∩

By a well known theorem about Euler obstructions (see [25, Sec. 3]),

EuMVN (x)=EuS(x).

Now, suppose we restrict the resolution φ˜ to S.

Lemma 5.14. φ˜ is a resolution of S such that the preimage of x is a rational curve normal with |S self intersection (N 1). − −

58 Proof. Let E be the preimage of x. Note that E is the proper transform of a line in the camera 1 plane of the i-th camera. To compute the self intersection of E in S˜ = φ˜− (S) consider the following embeddings: i j .E ֒ S˜ ֒ P˜3 −→ −→ By the Whitney sum formula, we have

˜ (ji) (c(N ˜))=(ji) c(N ˜3 ) φ∗( P2N ( 1)). ∗ E/S ∗ E/P ∩ O −

˜ ˜3 As we have already computed φ∗( P2N (1)) A•(P ), we just have to calculate (ji) c(N ˜3 ). O ∈ ∗ E/P By intersecting E with the generators of A2(P˜3) we ﬁnd

2 2 [E]= h + Qi + h Tij. j=i X6 3 Using this identity together with our presentation of A•(P˜ ) gives

3 (ji) c(N ˜3 )=[E] (N 1)h . ∗ E/P − −

Plugging everything into the Whitney sum formula shows that the degree of c(N ) is (N 1), E/S − − which completes the proof.

We now show that this self intersection number determines the Euler obstruction EuS(x).

Lemma 5.15. With x S the isolated singularity as above, Eu (x) = 3 N. ∈ S − Proof. Recall ( [56]) that a singularity germ (X,x) is taut if the analytic type of (X,x) is determined by the resolution graph of some resolution of singularities. By [56, 2.2] the vertex of the cone over the rational normal curve with degree n is taut. Let us denote this singularity by (Xn, 0). Since this singularity has a resolution in which the exceptional divisor is a P1 with self intersection n, − the resolution graph is a single vertex with weight (0, n). It follows that any singularity with this − resolution graph is analytically equivalent to (Xn, 0).

In particular, by lemma 5.14, (S,x) is analytically equivalent to (XN 1, 0) so the Euler ob- − struction EuS(x) is equal to the Euler obstruction EuXN−1 (0). By [8, 3.17], the latter is equal to 3 N. −

59 In conclusion, Eu (x) = 3 N, so equation 5.3 becomes MVN −

3 M M 1 φ˜ (c(P˜ )) = c (MVN )+(N 1)c (P ) ∗ −

3 M By plugging in our calculation of φ˜ (c(P˜ )) we obtain c (MVN ). ∗

Theorem 5.16. The Chern-Mather class of the multiview variety of N cameras in general position is 3 M i=0 ci (MVN ) where

P cM (MV ) = 4 + 4N 2N 2 + 2 N + 2 N • 0 N − 3 2 cM (MV ) = 7N N 2 +(N 4) N • 1 N − − 2 cM (MV ) = 4N 2 2 N 6 N 2N • 2 N − 3 − 2 − cM (MV )= N 3 (4 + N ) N N 2 N • 3 N − 2 − − 3 and cM (MV )= cM (MV ) [P2N i]. i N N ∩ − R 5.5.3 The ED degree of the multiview variety

As a corollary of theorem 5.16, we can compute the Euclidean distance degree of MVN .

Theorem 5.17. The ED degree of the multiview variety of N cameras in general position is equal to ED(MV ) = 6N 3 15N 2 + 11N 4. N − −

Proof. We can use the formula in [8] to express the sum of the polar degrees of MVN in terms of the Chern-Mather classes. Using this formula gives:

δ (MV ) = 6N 3 15N 2 + 11N 4. i N − − X Now, by the proof of [35, 6.11], if X is an aﬃne cone, then the ED degree of Xv is equal to the sum of the polar classes of Xv for a general translate Xv of X.

Suppose MVN is the multiview variety associated to the camera matrices P1,...,PN . Recall that MV P2N is the projective closure of a subvariety of C2N which we will call X. Let N ⊂

60 2N (v1,v2,...,v2N 1,v2N ) C be a vector. We will now show that Xv is multiview variety associ- − ∈ ated to a diﬀerent collection of cameras. Indeed, let Mi be the matrix

1 0 v2i 1 − M = 0 1 v i  2i  0 0 1   for 1 i N. Then, the variety X is the multiview variety associated to the cameras M P for ≤ ≤ v i · i 1 i N. ≤ ≤ In conclusion, there exists a general conﬁguration of cameras such that the ED degree of the associated multiview variety MVN is equal to the sum of the polar classes of MVN .

61 CHAPTER 6

TRITANGENT PLANES TO SPACE SEXTICS: THE ALGEBRAIC AND TROPICAL STORIES

This chapter is joint work with Yoav Len.

6.1 Introduction

In this chapter, we study tritangent planes to general sextic curves in three dimensional projective space (which, in particular, are not hyperelliptic). By general, we mean that the curve is the intersection of a smooth quadric and a smooth cubic. A plane in space is determined by three parameters, and when chosen generically, it meets a sextic in six points. Requiring that two contact points coincide to form a tangent imposes a single condition on the parameters. Therefore, a ﬁnite number of planes are expected to be tangent to the curve at three points. Making this argument precise and ﬁnding the exact number of tritangent planes is more subtle, and dates back to the mid-19th century with the work of Clebsch [28]. Sixty years later, an understanding of these tritangents was the impetus and principle goal for Coble in Algebraic Geometry and Theta Functions [29] The tritangent planes to a sextic are closely related with other classical problems such as the 27 lines on a cubic surface [49, Chapter V.4] and the 28 bitangents to a quartic curve [33, Chapter 6]. The projection from a general point of a cubic surface is a double cover of a plane branched along a quartic. The image of each of the 27 lines is bitangent to this quartic, with an additional bitangent given by blowing up the indeterminacy locus of the projection. Quite similarly, a del Pezzo surface of degree one forms a double cover of a quadric cone, branched along a smooth sextic of genus 4. The ( 1)-curves are mapped to conics, each meeting the sextic in three points, and the − planes containing these conics are tritangent planes, i.e., they have intersection multiplicity two at each of the three points. In a lecture given by Arnold at his 60th birthday conference at the Fields Institute, he referred to this as one of his mathematical trinities ( [12]). Other examples of trinities are the excep-

62 tional Lie algebras E6,E7,E8, the rings R, C, H, and the three polytopes tetrahedron, cube, and dodecahedron.

The next dream I want to present is an even more fantastic set of theorems and conjec- tures. Here I also have no theory and actually the ideas form a kind of religion rather than mathematics. The key observation is that in mathematics one encounters many trinities... I mean the existence of some functorial constructions connecting diﬀerent trinities. The knowledge of the existence of these diagrams provides some new conjec- tures which might turn to be true theorems... I have heard from John MacKay that the straight lines on a cubical surface, the tangents of a quartic plane curve, and the

tritangent planes of a canonic sextic curve of genus 4 form a trinity parallel to E6, E7

and E8.

Our interest in this chapter stems from two variations of the classical problem: the real case and the tropical case. In the case of a real space sextic, one may ask how many of the tritangent planes are real. In Sect. 6.3, we appeal to the theory of real theta characteristics to show that the answer depends on the topology of the real curve: the number of connected components and how they are arranged on the Riemann surface of the complex curve. In some cases, all the tritangents may be real, but their three points of tangency may include a complex-conjugate pair (Theorem 6.5). We continue to explore this phenomenon through explicit examples. We reexamine a construction of Arnold Emch, in which he claimed to ﬁnd 120 tritangents, and show that he over counted.

Theorem 6.6. Emch’s curve has only 108 planes tritangent at 3 real points.

On the other extreme, we construct a real space sextic with only one connected component and ﬁnd its 8 real tritangent planes. In Sect. 6.4 we set up a tropical formulation of the problem. Once the notion of a tropical tritangent plane is established, it is natural to ask how many such planes are carried by a tropical sextic curve. This is a natural sequel to earlier tropical counting problems, such as the number of lines on a tropical cubic surface [75], and the number of bitangents to a tropical plane quartic [15,26].

Theorem 6.17. A smooth tropical sextic curve Γ in R3 has at most 15 classes of tritangent planes. If it is the tropicalization of a sextic C on a smooth quadric in P3, then it has exactly 15 equivalence classes of tritangent planes.

63 In (Lemma 6.20) we show that the question can actually be replaced by the simpler problem of counting tritangents to tropical curves of bi-degree (3, 3) in tropical P1 P1. This result paves the × way for a computational study of tropical tritangents.

6.2 Algebraic space sextics

Throughout this section, we work over an algebraically closed field k of characteristic different from 2. For simplicity, the reader may assume that the field is C. Let C P3 be the intersection of a quadric and a cubic surface. Such a curve is a smooth ⊂ canonical sextic [49, Chapter IV, Prop. 6.3]. The intersection of a hyperplane with C is a divisor of degree 6 and rank 3, and is the canonical divisor KC of C. In particular, the genus of C is 4. It follows that whenever H is tangent to C at 3 points, those points form a divisor D such that 2D K . ≃ C Definition 6.1. A divisor class [D] with the property that 2D K is called a theta characteristic. ≃ C A theta characteristic is said to be odd or even according with the parity of dim H0(C,D).

Theorem 6.2. Let C be the sextic obtained from the intersection of a smooth cubic and a smooth quadric in P3. Then it has 120 tritangent planes, in bijection with its odd theta characteristics.

Proof. Let D be a theta characteristic of C obtained from a hyperplane section, and set h := dim H0(C,D). We will show that h = 1. To begin with, Cliﬀord’s theorem [11, Chapter 3.1] implies that h is strictly smaller than 3. As D is obtained from intersecting a curve with a plane, it is eﬀective, so h also cannot be 0. To see that h cannot be 2, recall the geometric version of the Riemann–Roch theorem: in the canonical embedding, the support of a divisor of degree d with h global sections spans a subspace of dimension d h. It follows that h = 2 if and only if the contact − points are co-linear. To see that this is impossible, let Q be the quadric containing C. Since Q is smooth, it is isomorphic to P1 P1. The intersection of Q with a plane H is a (1, 1)-curve on × Q. Therefore, if it contains a line of one of the rulings, it also contains a line in the other. In particular, H intersects C at points not in the support of D and is therefore not a tritangent.

On the other hand, given an odd theta characteristic p1 + p2 + p3, a plane H through p1,p2,p3 intersects C at a divisor of the form p1 + p2 + p3 + q1 + q2 + q3 for some points q1,q2,q3. Since p1 + p2 + p3 + q1 + q2 + q3 and 2(p1 + p2 + p3) are both canonical, we get an equivalence of divisors

64 p + p + p q + q + q . The rank of p + p + p is zero by Cliﬀord’s theorem and the fact that 1 2 3 ≃ 1 2 3 1 2 3 it is an odd theta characteristic. It follows that these divisors are, in fact, equal. We conclude that p1 + p2 + p3 + q1 + q2 + q3 = 2p1 + 2p2 + 2p3. In other words, H is tritangent to C at p1,p2,p3. We conclude that tritangent planes are in bijection with the odd theta characteristics of C. Itis well known [64] that the number of odd characteristics of a curve of genus g is 2g 1(2g 1), which − − is 120 in this case.

A canonical sextic does have two classes of co-linear divisors of degree 3. Indeed, those correspond to intersections of the sextic with the rulings of the ambient quadratic surface. However, as seen in the proof above, such a divisor is never obtained as the intersection of a hyperplane with the curve.

Remark 6.3. A smooth quadratic surface in P3 is isomorphic to P1 P1 via the Segre embedding [58, × Lemma 3.31]. Under this isomorphism, the sextic corresponds to a curve of bi-degree (3, 3) on P1 P1, and a tritangent plane corresponds to a tritangent (1, 1)-curve. It follows that a (3, 3)- × curve on a quadratic surface has 120 tritangent (1, 1)-lines as well. This result can also be deduced directly, similarly to the proof of Theorem 6.2.

Our initial interest in tritangent planes came from studying tritangent planes to Bring’s curve, as part of the apprenticeship workshop at the Fields Institute [73, Problem 4 on Curves]. Bring’s curve is a space sextic, traditionally written in supernumerary coordinates by considering a special plane in P4. In particular, it is the intersection of the quadric given by x2 + y2 + z2 + t2 + u2 = 0 and the cubic given by x3 + y3 + z3 + t3 + u3 = 0 in the plane x + y + z + t + u = 0. Edge [36] found equations for all 120 tritangent planes to Bring’s curve, which appear in two flavors. Type (i) tritangent planes are determined by three stalls of the curve. These are the points at which the osculating plane has order of contact higher than expected. In this case, the general point on the curve has order 3 contact with the osculating plane, and the stall points have order 4. Plücker’s formulas for space curves tell us that there are exactly 60 stalls on Bring’s curve. Let α,β,γ be the three distinct roots of θ3 + 2θ2 + 3θ + 4 = 0. One can check that each of the equations γt βu = 0, αu γz = 0, βz αt = 0 defines a plane which is tritangent to our − − − curve and contains the tangent line at the stall point (1, 1,α,β,γ). The rest of the type (i) planes are given by replacing (z,t,u) with any of the 60 ordered triples in x,y,z,t,u . Each of these { }

65 contains the tangent line at a stall point given by an appropriate permutation of the coordinates of [1 : 1 : α : β : γ]. The construction yields three tritangent planes through each of the 60 stall points of the curve, and every plane contains three stall points. In other words, the containment relation between type (i) tritangents and stalls determines the edges of a bipartite graph B(m, 60) such that every vertex has valence 3. Thus m = 60, so we see that there are 60 such tritangent planes. The type (ii) tritangent planes each contain exactly one stall point. One of them is given by

(α 1)(α + 4)z +(β 1)(β + 4)t +(γ 1)(γ + 4)u = 0, − − − and the rest are obtained, again, by replacing (z,t,u) with ordered triples in x,y,z,t,u . This is { } summarized in the theorem.

Theorem 6.4 ( [36]). Bring’s curve has 60 tritangent planes of type (i), and 60 tritangent planes of type (ii) with equations as above.

6.3 Tritangents to real space curves

In this section, we restrict ourselves to smooth curves that are deﬁned over the real numbers with non-empty real part. As is known, the real part of a curve consists of a disjoint union of ovals, where by oval we mean a simple closed loop. For a curve of genus g, the number of these ovals cannot exceed g + 1. See [22, 76] for a nice introduction to real algebraic geometry. We say that a tritangent plane is real if it is deﬁned over the reals, and totally-real if in addition the tangency points are all real. For a real tritangent that is not totally-real, the tangency points consist of a real point and a pair of complex conjugate points. For a smooth sextic on a smooth quadric in P3, real tritangent planes are in bijection with real odd theta characteristics. Their number is given as follows.

Proposition 6.5 ( [55]). Let C be a real curve of genus g, and assume that its real part C(R) consists of s> 0 ovals.

1. If C(R) separates C, then there are 2g 1(2s 1+1) real even theta characteristics and 2g 1(2s 1 − − − − − 1) real odd ones.

g+s 2 g+s 2 2. If C(R) doesn’t separate C, then there are 2 − real even and 2 − real odd theta characteristics.

66 N

Figure 6.1: Union of lines and a smooth cubic. Figure 6.2: Real sextic with ﬁve ovals.

A real curve of genus g with exactly g + 1 ovals is referred to as an M-curve. Any disjoint union of g + 1 cycles on a Riemann surface of genus g separates the surface, so an M-curve always corresponds to the first case of Proposition 6.5. In particular, a canonical space sextic with 5 ovals has 120 real tritangent planes. The question remains, how many of them are totally-real? In [39], Emch claimed that the tritangents of a real space sextic with 5 ovals are all totally-real, and constructed an example of such a curve and its tritangents. However, as we will see below, several of the planes were over counted, and only 108 of its tritangents are totally-real. We are not aware of any previous literature that has addressed this issue. We begin by considering a union of three lines in P2 so that they bound an equilateral triangle with incenter at the origin. For instance, if we choose one line to be of the form x = a, we find 1 that the other lines should have slope ± . Choosing a = √3 yields p(x,y)=(x + √3)(x y√3 √3 − − − 3)(x + y√3 3) and V (p(x,y)) A2 is our union of lines. − ⊂ The set of points (x,y) p(x,y) = 2 is a smooth cubic curve with four real branches (one of { | } which is an oval bounded by our triangle). The polynomial c(x, y, z) := p(x,y) 2 has zeros along − a cubic cylinder in A3. A sphere centered at the origin of sufficiently large radius meets each of the components of the cubic, and it meets the central component twice. Therefore, the intersection of the cubic surface with a sphere yields a space sextic with five ovals. We refer to the top and bottom ovals as N and S, and the other three as O1,O2,O3.

67 Theorem 6.6. The real sextic space curve determined by c(x, y, z) = 0 and x2 + y2 + z2 = 25 has 108 totally-real tritangent planes.

Proof. We break the proof into parts based on the type of tritangent plane. There are tritangents which touch three distinct ovals, tritangents which touch an oval twice and another oval once, and tritangents which touch a single oval three times. We label them (1, 1, 1), (2, 1), and (3)-tritangents respectively.

80 (1, 1, 1)-tritangents. Given any three ovals of the curve, there exist 23 = 8 classes of planes which separate them. That is, a given plane has some of the ovals “above” it, and some “below”. Such a plane can be moved so that it touches the three ovals each at one point in a unique way (For an analogue of this in the plane, consider two general non-concentric ellipses 5 and their four bitangents.) This yields 8 3 = 80 tritangents, and there are clearly no other tritangent planes meeting each of three ovals once.

12+18 (2, 1)-tritangents. For each Oi, there are four tritangents which touch it twice. To ﬁnd

them, consider the plane tangent to Oi at its northernmost point and southernmost point.

The plane can be rotated so that it keeps two tangency points to Oi. As it rotates, it meets the other two O each once yielding two tritangents for a total of 3 2 = 6 such tritangents. j · The projection of these two tritangents to the xy-plane is pictured in Figure 6.3. Similarly, for N (resp. S), there are nine tritangents which touch it twice. To see them, pick a

side of the triangle of N (resp. S), and consider the opposing Oi. There is a tritangent which

touches N (resp. S) at two points along this side and touches Oi at its northernmost point,

and similarly one which touches the Oi’s southernmost point. Finally, there is a tritangent which touches the opposing point of S (resp. N). This yields 9 2 = 18 tritangents. · Clearly there are no additional (2, 1)-tritangents meeting N,S twice. We now show there are

no additional (2, 1)-tritangents meeting Oi twice. Observe that the oval Oi has two reflectional symmmetries, one through the “equator” and one through the great circle determined by the northernmost and southernmost points. If p O is a point which is not fixed by either ∈ i reflection, then the images under reflection – denoted p′ and p′′ each share a tangent plane

to Oi with p, that is, the tangent line TpOi to Oi at p intersects Tp′ Oi and Tp′′ Oi. The two

planes determined by these lines are the only bitangent planes to Oi at p. If p′ is given by reﬂecting p through the xy-plane, then the corresponding bitangent is a tritangent only if the projection (Figure 6.3) is a bitangent line, and it is apparent from the picture that we have

already claimed all these. If p′′ is the other reﬂection, then the bitangent to p and p′′ cuts

out a circle on the sphere which is contained in Oi. Therefore it cannot be tangent at a point on another oval.

68 Figure 6.3: Two (2, 1)-tritangents projected to xy-plane.

4 (3)-tritangents. N has three maxima with respect to height in the z-direction. There is a plane which touches the oval at these three points. Similarly, it has three minima, and there is another tritangent plane there. The same is true for S. We thus have 2 2 = 4 four more · tritangent planes. It is easy to see that there are no more (3)-tritangent planes meeting N or S only. A tritangent

plane also cannot meet Oi only as, by symmetry, such a plane would have to touch Oi at either its northernmost or southernmost point, but there are not other points sharing a bitangent with either of these.

We have shown that Emch’s curve has fewer totally-real tritangents than was previously thought. A natural question is thus reopened.

Question 6.7. Does there exist a canonically embedded real space sextic with 120 totally-real tritangent planes?

We now consider a curve with signiﬁcantly fewer totally-real tritangent planes. Let C be the sextic determined by the intersection of the sphere S2 deﬁned by

x2 + y2 + z2 = 1

69 Figure 6.4: The intersection of a cubic and a Figure 6.5: A real sextic curve with a single quadric yields a sextic. connected component.

with the “Clebsch diagonal cubic” S3 deﬁned by

81(x3 + y3 + z3) 189(x2y + x2z + xy2 + y2z + xz2 + yz2) + 54xyz − + 126(xy + xz + yz) 9(x2 + y2 + z2) 9(x + y + z) + 1 = 0. − − This cubic surface has a threefold rotational symmetry about the axis x = y = z. In Figure 6.5, 2π this corresponds to the 3 rotation about the “north/south poles”, which we will call pN and pS. Proposition 6.5 implies that since C has real points, it has eight real tritangent planes. As the following theorem shows, these planes exist and are all totally-real.

Theorem 6.8. The curve C has exactly 8 totally-real tritangent planes.

Proof. Let q denote a point on C which minimizes the distance to pN . This point lies on a circle in 3 R which is the intersection of S2 with a sphere centered at pN of radius d(pN ,q). By the threefold rotational symmetry, there are at least three distinct points at which C touches the circle. Thus, the plane containing this circle is tritangent to C and there are exactly three tangency points. The same argument for pS gives a second tritangent plane to C.

Observe that C has a reﬂectional symmetry through the plane determined by pN , pS, and q.

The three points associated to any tritangent to C must lie on a circle on S2 which cannot cross C. By the reﬂectional symmetry of C, this circle must meet C at one of the points q from the two

70 known tritangents. Such a circle, of a sufficiently small radius, meets C at no other points. With a sufficiently large radius, it meets C transversely at multiple points. Thus, there are circles on either side of C, which touch but don’t cross the curve at some point other than q. Since the total intersection number of the two curves cannot exceed 3 2 = 6, and by the reflectional symmetry, · this circle is tangent at exactly three points. Hence, we find one additional circle for each of the six points q, for a total of 8 tritangents planes. By Proposition 6.2, this is the maximum possible.

6.4 Tropical space sextics

In this section, we consider tropical space sextics of genus 4, and show that they satisfy enumerative properties that are analogous to the ones presented above. We begin with a brief overview of several topics regarding tropical curves. We mostly focus on notions that are necessary for deﬁning and studying tropical tritangent planes. The interested reader may ﬁnd a more thorough treatment in [59].

Deﬁnition 6.9. A tropical curve is a metric graph Γ embedded in Rn, together with an integer weight function on the edges, such that The direction vector of each edge is rational. • At each vertex, the weighted sum of the primitive integral vectors of the edges around the • vertex is zero.

The genus of a tropical curve is the ﬁrst Betti number dim H1(Γ, Z) of the graph.

We assume throughout that the weights on the edges are all one.

Deﬁnition 6.10. A tropical curve is of degree d if it has d inﬁnite ends in each of the directions e , e ,..., e ,e + e + + e . A plane curve is of bi-degree (d ,d ) if it has d ends in each − 1 − 2 − n 1 2 ··· n 1 2 1 of the directions e , e , and d ends in the directions e , e . 2 − 2 2 1 − 1 Example 6.11. The graph in Figure 6.6 is a tropical plane curve of degree 3 and genus 1.

Deﬁnition 6.12. A tropical plane in R3 is a two dimensional polyhedral complex with a unique vertex v, whose 1-skeleton consists of the rays v + R 0( e1),v + R 0( e2),v + R 0( e3), and v + R 0(e1 + e2 + e3). The maximal faces are ≥ − ≥ − ≥ − ≥ the cones generated by each pair of rays. In other words, it is a translation of the 2-skeleton of the fan of the toric variety P3.

71 Figure 6.6: A tropical elliptic curve.

More generally, a tropical variety is a balanced polyhedral complex in Rn, in the sense of [59, Definition 3.3.1]. Tropical hypersurfaces (namely tropical varieties of codimension 1) are simply constructed by taking the dual complex of a subdivision of a polytope with integer vertices. The hypersurface is tropically smooth if the subdivision is a unimodular triangulation. For the rest of this section, we assume that k is an algebraically closed field, endowed with a non-trivial non-archimedean valuation ν. For simplicity, we may choose k to be the field of Puiseux series over C, consisting of all elements of the form

k k+1 x = a t n + a t n + k · k+1 · ··· for all choices of k Z,n N, and coeﬃcients a C. The valuation is given by ν(x)= k . ∈ ∈ i ∈ n Let X be a variety in (k )n. The tropicalization map trop : X(K) Rn is deﬁned by ∗ → trop(x ,x ,...,x )=( ν(x ), ν(x ),..., ν(x )). 1 2 n − 1 − 2 − n The tropicalization of X is the closure in Rn of trop(X(k)). The reader will be relieved to know that the tropicalization of a variety is, indeed, a tropical variety of the same dimension. The tropicalization of a generic curve of degree d (resp. bi-degree

(d1,d2)) is a tropical curve of degree d (resp. bi-degree (d1,d2)). Similarly, the tropicalization of a 3 3 plane in (k∗) is a tropical plane in R .

Example 6.13. The tropicalization of the degree 3 plane curve

f = t + x + y + xy + t x2 + t y2 + t2 x2y + t2 xy2 + t4 x3 + t4 y3 · · · · · · is the tropical curve of degree 3 appearing in Figure 6.6. The tropicalization of the curve of bi-degree (1, 2) g =1+ x + y + t xy + t3 xy2 + t3 y2 · · ·

72 Figure 6.7: A tropical P1 P1 curve of bi-degree (1,2). × is the tropical curve of bi-degree (1, 2) depicted in Figure 6.7.

A common diﬃculty in tropical geometry is that occasionally, tropical varieties intersect at higher than expected dimension. This can happen even when the tropical varieties in question are tropicalizations of algebraic varieties that do intersect in the expected dimension. One way to address this issue is to “force” the varieties to intersect properly, using the so-called stable intersection. Colloquially, this is done by generically perturbing the tropical varieties, and taking the limit of their intersection as the perturbations tend to zero. n More precisely, whenever cells σ1,σ2 of tropical varieties Σ1, Σ2 span R , their set-theoretic intersection will be a cell in the stable intersection. To assign a weight to this cell, consider the n n lattices N1,N2 obtained by intersection σ1,σ2 with the lattice Z in R . The weight given to the cell is m(σ ) m(σ ) [Zn : N + N ], 1 · 2 · 1 2 where m(σi) is the weight of each cell σi.

Example 6.14. Consider the two curves depicted in Figure 6.8. A basis for the ray of the blue curve at the intersection point is given by the primitive vector (1, 1), whereas a basis for the ray of the red curve is given by (1, 1). The multiplicity of the point of intersection is therefore − 1 1 det = 2. 1 1 − This is consistent with tropical Bezout’s theorem, since this is an intersection of a line with a (1, 1)-curve. The two curves in Figure 6.9 don’t intersect properly. However, by slightly perturbing the red curve, the two curves intersect at two points with multiplicity 1. Taking the limit as the red curve returns to its original position, we see that the stable intersection has multiplicity 2.

73 Figure 6.8: Two tropical curves meeting with Figure 6.9: Tropical curves intersecting stably multiplicity 2. at two points.

Deﬁnition 6.15. Two tropical varieties are tangent at a point q if their intersection at q has weight at least 2, or if q is in the interior of a bounded segment of their set-theoretic intersection. They are tritangent to each other if they are tangent at three disjoint places (either points or segments) counted with multiplicity. Two tritangents are equivalent if the tangency points are linearly equivalent divisors.

In particular, tritangent varieties may meet at three places with multiplicity 2 each, at two places with multiplicity 4 and 2, or at one place with multiplicity 6. See Figure 6.11 for various examples of curves that are tritangent to each other. When Γ is the tropicalization of a curve C, then any tangent of C tropicalizes to a tangent of Γ, and in particular a tritangent tropicalizes to a tritangent. This can be deduced, for example, from [65, Theorem 6.4].

6.5 Tropical divisors, theta characteristics and tritangent planes

Divisors on tropical curves are defined analogously to algebraic curves. A divisor D on Γ is a finite formal sum D = a p + a p + + a p , 1 1 2 2 ··· k k where each a is an integer, and each p is a point of the curve. The degree of D is a +a + +a , i i 1 2 ··· k and we say that D has a chips at p . The divisor is said to be effective if a 0 for every i. In i i i ≥ analogy with the algebraic case, there is a a suitable equivalence relation between divisors, and a notion of rank which, roughly speaking, reflects the dimension in which the divisor moves. The curve has a canonical divisor class KΓ which fits in a tropical Riemann–Roch theorem

r(D) r(K D) = deg(D) g + 1, − Γ − −

74 where r is the rank of a divisor, and g the genus of the tropical curve ( [16, Theorem 1.12]). See [14] for a lucid introduction to tropical divisor theory. Now, a tropical theta characteristic on a tropical curve Γ is defined in exactly the same way as algebraic theta characteristic. Namely, it is a divisor class [D] such that 2D K . Since ≃ Γ the Jacobian of a tropical curve of genus g is isomorphic to a g-dimensional real torus Rg/Zg ( [13, Theorem 3.4] and [23, Section 5]), and theta characteristics are in bijection with its 2-torsion points, there are 2g theta characteristics. One of them is non-effective and the rest are effective. They are easily computed via the following algorithm, introduced by Zharkov [77]. To get an effective theta characteristic, fix a cycle γ in Γ. At every point p that locally maximizes the distance from γ, place a 1 chips at p, where a is the number of incoming edges at p from the − direction of γ. The process is often described pictorially as follows. A fire spreads along the graph at equal speed away from γ. If a is the number of incoming fires at a point p, we place a 1 chips − at that point. To obtain the unique non-effective theta characteristic, repeat the same process, but replace γ with the set of vertices of the graph, and place a negative chip at each vertex.

Example 6.16. Let Γ be the curve in Figure 6.10 (where the inﬁnite ends are omitted). As the genus is 2, we expect the theta characteristics to have degree 1. For the picture on the left, the middle of the bottom horizontal edge is the unique local maximum from the chosen cycle (marked with a red cycle). The corresponding theta characteristic has a single chip at that point. For the picture on the right, the middle of each horizontal edge locally maximizes the distance from the vertices. The non-eﬀective theta characteristic of this curve therefore consists of a negative chip at each of the three vertices, and a chip at the middle of each horizontal edge. Each of these divisors, when multiplied by two is equivalent to the canonical divisor, so they are half canonical.

1 1 − −

Figure 6.10: Two theta characteristics on a curve of genus 2.

Let Γ be a tropical curve of degree 6 and genus 4 in R3. Its stable intersection with a tropical plane Π is an eﬀective divisor of degree 6. We claim that its rank is 3. Indeed, we can ﬁnd

75 a tropical plane through any three general points, and any pair of divisors obtained this way is linearly equivalent. By the tropical Riemann–Roch theorem, a divisor of degree 6 and rank 3 has to be equivalent to the canonical divisor. Consequently, every tritangent plane gives rise to an effective theta characteristic on Γ. By definition, non-equivalent tritangent planes correspond to different theta characteristics. It follows that the number of equivalence classes of tritangent planes is bounded above by the number of effective theta characteristics which is 24 1 = 15. − Theorem 6.17. A smooth tropical sextic curve Γ in R3 has at most 15 classes of tritangent planes. If it is the tropicalization of a sextic C on a smooth quadric in P3, then it has exactly 15 equivalence classes of tritangent planes.

Proof. The first statement was shown in the discussion above. By Theorem 6.2, the curve C has 120 tritangent planes. If H is tritangent to C, then trop(H) is tritangent to Γ. Moreover, by [53, Theorem 1.1], each tropical effective theta characteristic of Γ is the tropicalization of 8 odd theta characteristics of C. Since different theta characteristics correspond to non-equivalent tritangent planes, there are 15 distinct classes of planes tritangent to Γ.

Remark 6.18. It is quite possible for a tropical sextic to have an inﬁnite continuous family of tritangent planes. However, the tangency points of such a family will consist of linearly equivalent divisors, and as such the corresponding tritangent planes are equivalent. If the sextic is the tropicalization of an algebraic sextic, then each equivalence consists of 8 tritangent planes (counted with multiplicity) that can be lifted to tritangent planes of the algebraic sextic.

The proof above relies on the fact that the given tropical curve arises as the tropicalization of an algebraic curve. However, we expect the result to be true in general.

Conjecture 6.19. Every tropical sextic curve of genus 4 in R3 has exactly 15 equivalence classes of tritangent planes.

We now wish to explore the tropical analogue of the relation between quadric surfaces and P1 P1, as described in Remark 6.3. To examine the analogous statement in tropical geometry, recall × that a smooth tropical quadric in R3 is dual to a unimodular triangulation of the 3-simplex with vertices (0, 0, 0), (2, 0, 0), (0, 2, 0), (0, 0, 2). By the proof of [59, Theorem 4.5.8], such a triangulation has a unique interior edge, corresponding to a unique bounded face of the quadric. This face can be seen as a model for tropical P1 P1. More precisely, ×

76 Lemma 6.20. Let Σ be a tropical smooth quadric surface in R3, and ﬁx a rectangle R in R2. Then there is an aﬃne linear map from R onto a parallelogram in the bounded face of Σ inducing a bijection between curves of bi-degree (d,d) in R2 whose bounded edges are all contained in R, and curves of degree 2d in R3 that are contained in Σ, and their bounded edges are contained in the parallelogram.

Proof. Denote by F the unique bounded face of Σ. By [59, Theorem 4.5.8], there are two tropical lines through each point of F that are fully contained in Σ. The bounded edge of each is fully contained in F , and is parallel to one of two directions, which we denote u1 and u2. These directions are determined by F , and do not depend on the point. Every ray in F that is parallel to ui (for i = 1, 2) can be extended past the boundary of F by attaching infinite ends in two of the directions e , e , e ,e (where e = e + e + e ), and a ray parallel to u can be extended {− 1 − 2 − 3 0} 0 1 2 3 − i by attaching ends in the two remaining directions. Let ϕ : R F be the affine linear map that sends a vertex of R to p, and the two adjacent → vertices to p + λu1,p + λu2 for some fixed point p of F , and a small enough λ so that the image is contained in F . Let Γ be a (d,d)-curve Γ in R2 whose bounded edges are contained in R. Each of the infinite ends is mapped by ϕ to rays that are parallel to u or u . By extending all the ± 1 ± 2 rays emanating from ϕ(Γ R) according to the discussion above, we get a curve of degree 2d in R3 ∩ that is contained in Σ.

The lemma suggests an alternative approach for proving Conjecture 6.19.

Corollary 6.21. For tropical sextics whose bounded edges are contained in the bounded face of a smooth tropical quadric, Conjecture 6.19 is equivalent to the statement that every (3, 3)-curve in R2 has 15 classes of tritangent (1, 1)-curves.

Proof. Let Γ be a (3, 3)-curve in R2, mapping to a sextic in R3 by ϕ as in Lemma 6.20. Every (1, 1)-curve that is tritangent to Γ maps to a conic curve in R3 that is tritangent to ϕ(Γ) and contained in Σ. By construction, the conic curve is not contained in a tropical line or in any of the standard planes in R3. Therefore, we can ﬁnd 3 general points on it that span a unique tropical plane. This plane contains the conic curve and is tritangent to ϕ(Γ).

Example 6.22. The Figure below shows 15 equivalence classes of tritangent (1, 1)-curves to a tropical (3, 3)-curve in R2. By Corollary 6.21, this curve corresponds to a tropical sextic in R3 reaching the

77 maximal number of tritangent planes. To ﬁnd each tritangent curve, we choose a non-trivial cycle in the graph, compute the corresponding theta characteristic via Zharkov’s algorithm, and ﬁnd a (1, 1)-curve through it.

2 2

Figure 6.11: 15 tritangents to a tropical sextic.

We stress that some of the odd theta characteristics, in fact, give rise to inﬁnitely many tritangent (1, 1)-curves, for instance the third tritangent in the second row in Figure 6.11 when counting from the top left. However, as the tangency points of these diﬀerent (1, 1)-curves are equivalent

78 divisors, they are all in the same equivalence class (cf. Remark 6.18).

79 CHAPTER 7

EQUATIONS AND TROPICALIZATION OF ENRIQUES SURFACES

This chapter is joint work with Barbara Bolognese and Joachim Jelisiejew.

7.1 Introduction

In the classiﬁcation of algebraic surfaces, Enriques surfaces comprise one of four types of minimal surfaces of Kodaira dimension 0. There are a number of surveys on Enriques surfaces. For those new to the theory, we recommend the excellent exposition found in [18] and [20], and for a more thorough treatment, the book [31]. Another recommended source is Dolgachev’s brief introduction to Enriques surfaces [34]. The ﬁrst Enriques surface was constructed in 1896 by Enriques himself [40] to answer negatively a question posed by Castelnuovo (1895):

Is every surface with pg = q = 0 rational?

(see Section 7.2 for the meaning of pq and q). Enriques’ original surface has a beautiful geometric construction: the normalization of a degree 6 surface in P3 with double lines given by the edges of a tetrahedron. Another construction, the Reye congruence, defined a few years earlier by Reye [68], was later proved by Fano [41] to be an Enriques surface. Since these first constructions, there have been many examples of Enriques surfaces, most often as quotients of K3 surfaces by a fixed- point-free involution. In [30], Cossec describes all birational models of Enriques surfaces given by complete linear systems. As we recall in Section 7.2, every Enriques surface has an unramified double cover given by a K3 surface. Often exploiting this double cover, topics of particular interest relate to lattice theory, moduli spaces and their compactifications, automorphism groups of Enriques surfaces, and Enriques surfaces in characteristic 2.

80 While there are many constructions of Enriques surfaces, none give explicit equations for an Enriques surface embedded in a projective space or in a more general toric variety. In this chapter we give explicit ideals for all Enriques surfaces.

Theorem 7.1. Let Y be the toric ﬁvefold of degree 16 in P11 that is obtained by taking the join of the Veronese surface in P5 with itself. The intersection of Y with a general linear subspace of codimension 3 is an Enriques surface, and every Enriques surface arises in this way.

By construction, the Enriques surface in Theorem 7.1 is arithmetically Cohen-Macaulay. Its homogeneous prime ideal in the polynomial ring with 12 variables is generated by the twelve binomial quadrics that deﬁne Y and three additional linear forms. Explicit code for producing this Enriques ideal in Macaulay 2 is given in Section 7.3. The contents of the chapter are as follows. In Section 7.2 we give some background about Enriques surfaces. Next, in Section 7.3, we exploit a classical construction to obtain an Enriques ideal in a codimension 3 linear space in P11 and prove Theorem 7.1.

7.2 Background

Apart from the code snippets, we work over an algebraically closed field k of characteristic zero. 1 2 An Enriques surface X is a smooth projective surface with q(X) := h (X, ) = 0, ω⊗ and OX X ≃OX ω , where ω = 2 Ω1 is the canonical bundle of X. Then it follows that X is minimal, X 6≃ OX X X 2 see [20], and pg(X) := Vh (X, X ) = 0. We note that Enriques surfaces are defined the same O way over any field of characteristic other than 2. By [19, Lemma 15.1] the Hodge diamond of an Enriques surface X is:

h0,0 1 h1,0 h0,1 0 0 h2,0 h1,1 h0,2 = 0 10 0 h2,1 h1,2 0 0 h2,2 1 . (7.1)

An Enriques surface admits an unramiﬁed double cover f : Y X, where Y is a K3 surface, → see [19, Lemma 15.1] or [20, Proposition VIII.17]. The Hodge diamond of Y is given by

81 h0,0 1 h1,0 h0,1 0 0 h2,0 h1,1 h0,2 = 1 20 1 h2,1 h1,2 0 0 h2,2 1 . (7.2)

In particular since Y is simply-connected, the fundamental group of an Enriques surface is Z/2Z, see [19, Section 15]. The cover Y X is in fact a quotient of Y by an involution σ, which → exchanges the two points of each ﬁber. Conversely, for a K3 surface Y with a ﬁxed-point-free involution σ the quotient Y/σ is an Enriques surface.

7.3 Enriques surfaces via K3 complete intersections in P5

In this section we construct Enriques surfaces via K3 surfaces in P5. Before we go into the details, we remark that one cannot hope for easy equations, for example an Enriques surface cannot be a hypersurface in P3.

Proposition 7.2. Let X PN be a smooth projective toric threefold and S = X H be a smooth ⊂ C ∩ hyperplane section. Then S is simply-connected. In particular it is not an Enriques surface.

Proof. Since X is a smooth and projective toric variety, it is simply connected by [43, 3.2]. Now a § homotopical version of Lefschetz’ theorem ( [17] see also [21, 2.3.10]) asserts that the fundamental groups of X H and X are isomorphic via the natural map. Thus S is simply connected. Now ∩ suppose S is an Enriques surface. Then it admits a non-trivial `etale double cover K S, thus it → is not simply connected, which is a contradiction.

We remark that this proof generalizes to other complete intersections inside smooth toric varieties, provided that intermediate complete intersections are smooth as well.

We now construct an Enriques surface from a K3 surface which is an intersection of quadrics in P5. We follow Beauville [20, Example VIII.18]. 5 Fix a projective space P with coordinates x0, x1, x2, y0, y1, y2. Consider the involution σ : P5 P5 given by σ(x ) = x and σ(y ) = y for i = 0, 1, 2. Then the ﬁxed point set is equal → i i i − i 2 2 to the union of P = V (y0,y1,y2) and P = V (x0,x1,x2).

82 Fix quadrics F C[x ,x ,x ] and G C[y ,y ,y ], where i = 0, 1, 2 and denote Q := F +G . i ∈ 0 1 2 i ∈ 0 1 2 i i i By their construction, these quadrics are fixed by σ. We henceforth choose Qi so that they give a complete intersection. Then S = SQ := V (Q0,Q1,Q2) is a surface and, by the Adjunction Formula, we have K = ( 6+2+2+2) = . It can also be shown that since the surface S is a complete S OS − OS intersection of quadrics in P5, it has h1( ) = 0, see [20, Lemma VIII.9]. Thus if S is smooth, then OS it is a K3 surface fixed under the involution σ. We will now formalize exactly which assumptions must be satisfied by the three quadrics to obtain a smooth Enriques surface.

Deﬁnition 7.3. Let Q = (Q ,Q ,Q ) be a triple of quadrics Q = F + G for F C[x ] and 0 1 2 i i i i ∈ i G C[y ] as before. We say that the quadrics Q are enriquogeneous if the following conditions i ∈ i are satisﬁed:

1. the forms Q =(Q0,Q1,Q2) are a complete intersection,

2. the surface S = V (Q0,Q1,Q2) is smooth,

3. the surface S = V (Q0,Q1,Q2) does not intersect the ﬁxed-point set of σ.

We note that the third condition is equivalent to F1,F2,F3 having no common zeros in C[x0,x1,x2] and Gi having no common zeros in C[y0,y1,y2], so it is open. We know that for a choice of enriquogeneous quadrics Q we obtain an Enriques surface as SQ/σ. The set of enriquogeneous quadrics is open inside (A6)6, so that a general choice of forms gives an Enriques surface. In [30], Cossec proves that every complex Enriques surface may be obtained as above if one allows Q not satisfying the smoothness condition, see also [74]. Notably, Lietdke recently proved that the same is true for Enriques surfaces over any characteristic [57]. To give some intuition for the complex result, let us prove that over the complex numbers these surfaces give at most a 10-dimensional space of Enriques surfaces.

Notice that each Qi is chosen from the same 12-dimensional aﬃne space and SQ depends only on their span, which is an element of Gr 3, C12 . This is a 27-dimensional variety. However, since we have ﬁxed σ, the quadrics Qi will give an isomorphic K3 surface (with an isomorphic involution) if we act on P5 by an automorphism that commutes with σ. Such automorphisms are given by block matrices in PGL(6) of the form

A 0 0 A C = or C = (7.3) 0 B B 0

83 where A and B are matrices in GL(3), up to scaling. Thus, the space of automorphisms preserving the σ-invariant quadrics has dimension 2 9 1 = 17. Modulo these automorphisms, we now have · − a 10-dimensional projective space of K3 surfaces with an involution. Note that the condition that Q be enriquogeneous is an open condition.

We now aim at making the Enriques surfaces obtained above as SQ/σ explicit. In other words we want to present them as embedded into a projective space. 5 The ﬁrst step is to identify the quotient of P by the involution σ. Let S = C[x0,x1,x2,y0,y1,y2] σ be the homogeneous coordinate ring. Then the quotient is Proj (S ) = Proj(C[xi,yiyj]). The

Enriques SQ is cut out of Proj (C[xi,yiyj]) by the quadrics Q, so that

SQ = Proj(C[xi,yiyj]/Q) . (7.4)

8 This does not give us an embedding into P , since the variables xi and yiyj have diﬀerent degrees. Rather we obtain an embedding into a weighted projective space P(13, 26). Therefore we replace

C[xi,yiyj] by the Veronese subalgebra

SQ Proj(C[x x ,y y ]/Q) . (7.5) ≃ i j i j

This algebra is generated by 12 elements xixj, yiyj for i,j = 0, 1, 2, so that SQ is embedded into a 11 8 P . The relations Q are linear in the variables xixj and yiyj, so that SQ is embedded into a P . Let us rephrase this geometrically. Consider the second Veronese re-embedding v : P5 P20. → 20 The coordinates of P are forms of degree two in xi and yi. The involution σ extends to an involution on P20 and this time the invariant coordinate ring is generated by the linear forms 11 corresponding to products xixj and yiyj. Therefore the quotient is embedded into P , which has coordinate ring corresponding to those 12 forms.

v P5 P20

π P11 (7.6) where π denotes the quotient by the involution σ. Then the image π(P5) is cut out by 12 binomial quadrics, which are the 6 usual equations between xixj and the 6 corresponding equations for yiyj.

It is the join of two Veronese surfaces which constitute its singular locus. Quadrics in C[xi,yi]

84 which have the form F + G for F C[x ] and G C[y ] correspond bijectively to linear forms i i i ∈ i i ∈ i on the above P11. A choice of enriquogeneous quadrics Q corresponds to a general choice of three 11 5 linear forms on P . We obtain the corresponding Enriques surface SQ as a linear section of π(P ). Summing up, we have the following chain of inclusions

V π(P5) π(P5) P(13, 26) P11 (7.7) ∩ ⊂ ⊂ ⊂ where V is a codimension three linear section. Note that although V π(P5) is a complete inter- ∩ section in π(P5), this is not contradictory to (a natural generalisation of) Proposition 7.2, because π(P5) is singular. Note also that suﬃciently ample embeddings of varieties are always cut out by quadrics, see [63, 70], so this suggests that our embedding is suﬃciently good.

Proof of Theorem 7.1. The surfaces obtained from enriquogeneous quadrics are arithmetically Cohen- Macaulay of degree 16 as they are linear sections of π(P5) possessing those properties. Every En- riques surface can be obtained by this procedure if one allows Q not satisfying the smoothness condition by [30].

Below we provide Macaulay2 code for obtaining the equations of SQ explicitly, using the above method. We could take any field as kk; we use a finite field to take random elements. kk = ZZ/1009; P5 = kk[x0,x1,x2,y0,y1,y2]; P11 = kk[z0,z1,z2,z3,z4,z5,z6,z7,z8,z9,z10,z11]; pii = map(P5, P11, {x0^2, x0*x1, x0*x2, x1^2, x1*x2, x2^2, y0^2, y0*y1, y0*y2, y1^2, y1*y2, y2^2});

We can verify that the kernel of pii is generated by 12 binomial quadrics and has degree 16. assert(kernel pii == ideal(z10^2-z9*z11, z8*z10-z7*z11, z8*z9-z7*z10, z8^2-z6*z11, z7*z8-z6*z10, z7^2-z6*z9, z4^2-z3*z5, z2*z4-z1*z5, z2*z3-z1*z4, z2^2-z0*z5, z1*z2-z0*z4, z1^2-z0*z3)) assert(degree kernel pii == 16)

Now we generate an Enriques from a random set of linear forms named linForms. To see the quadrics in P5 take pii(linForms).

85 Quadrics = random(P11^3, P11^(-1))

randomEnriques = (kernel pii) + ideal linForms

We now check whether randomEnriques is in fact an Enriques surface. Computationally it is much easier to check this for the associated K3 surface, since we need only check that K3 is a smooth surface (ﬁrst two assertions below) and that the involution is ﬁxed-point-free on K3 (last two assertions).

K3 = ideal pii(linForms) assert (dim K3 == 3) assert (dim saturate ideal singularLocus K3 == -1) assert (dim saturate (K3 + ideal(y0,y1,y2)) == -1) assert (dim saturate (K3 + ideal(x0,x1,x2)) == -1)

If the K3 passes all the assertions, then randomEnriques is an Enriques surface. Its ideal is given by 12 binomial quadrics listed above and three linear forms in P11.

Example 7.4. Over k = F1009 the choice of linForms = matrix{{2*z2+z6+5*z7+8*z11, 2*z0+8*z4+z9, 5*z1+4*z3+4*z5+6*z8}} in the above algorithm gives an Enriques surface.

Finally, we check that π(P5) is arithmetically Cohen-Macaulay. Using betti res kernel pii we obtain its Betti table.

1...... 12 16 6 . . . (7.8)   . . 36 96 100 48 9   The projective dimension of π(P5) (the number of columns) is equal to the codimension, thus π(P5) P11 is arithmetically Cohen-Macaulay, see [69, Section 10.2]. Therefore all its linear ⊂ sections are also arithmetically Cohen-Macaulay.

86 APPENDIX A

SOURCE CODE FOR MACAULAY2 PACKAGE FMPINTERSECTIONTHEORY

The package FMPIntersectionTheory for the computer algebra system Macaulay2 implements the main algorithm from chapter 2 as well as some the applications and ancillary functions exhibited in this thesis. In this appendix, the source code for the most important parts of the code base is included with supplementary remarks. First, we create a new type called ProjectiveScheme that knows some things about its intersection theory, for instance the chow ring of the ambient projective space and its fundamental class.

ProjectiveScheme = new Type of MutableHashTable globalAssignment ProjectiveScheme toString ProjectiveScheme := net ProjectiveScheme := X -> ( if hasAttribute(X,ReverseDictionary) then ( toString getAttribute(X,ReverseDictionary) ) else "a projective scheme" )

ProjectiveScheme#{Standard,AfterPrint} = X -> ( << concatenate(interpreterDepth:"o") << lineNumber << " : " << "a projective scheme in PP^" << dim(X.AmbientSpace) << " defined by " << X.Ideal << endl; ) projectiveScheme = method(TypicalValue => ProjectiveScheme, Options => { SuperScheme => null, -- a ProjectiveScheme containing the one we are defining -- If I is the ideal of SuperScheme in R, and we define our -- new scheme with J in R, we instead will use I+J AmbientSpace => null, -- the projective space where we will be computing MakeBaseOfLinearSystem => false -- if true, the ideal used to define the scheme -- should be made to have all terms of same degree

87 })

Now we deﬁne the constructor for the ProjectiveScheme type. projectiveScheme Ideal := opts -> I -> ( if opts.SuperScheme =!= null then ( I = trim (I + opts.SuperScheme.Ideal); ); if opts.MakeBaseOfLinearSystem then ( I = homogenate(I); ); R := ring I; N := 0; eqs := flatten entries gens I; P := if opts.SuperScheme =!= null then ( opts.SuperScheme.AmbientSpace ) else if opts.AmbientSpace =!= null then ( opts.AmbientSpace ) else ( N = #(flatten entries vars R) - 1; -- dim of proj(R) projectiveBundle N );

new ProjectiveScheme from { global Ideal => I, global BaseField => coefficientRing ring I, global CoordinateRing => quotient I, global Equations => eqs, global AmbientSpace => P, global Hyperplane => ( chern_1 (OO_P(1)) ), global dim => null, global degree => null, global codim => null, CycleClass => null } )

This function takes the ideal I and ﬁnds a set of generators which are all of the same degree. homogenated := I -> ( -- get a list of degrees of the generators of I and take the max gns := flatten entries gens I; degs := apply(gns, g -> (degree g)#0); maxDeg := max(degs);

88 -- test whether all degrees attain the max return all(degs, d -> d == maxDeg); ) homogenate := I -> ( if (homogenated I) then return I;

-- get list of generators and take max degree gns := flatten entries gens I; maxDeg := max(apply(gns, g -> (degree g)#0));

-- split the list into sublists by degree -- e.g. {z, xy2, x2y, x3+y3} -> {{}, {z}, {}, {xy2, x2y}, {x3+y3}} gLists := for i from 0 to maxDeg list ( select(gns, g -> (degree g)#0 == i) );

J := ideal ( vars ring I ); gs := for i to ( (length gLists)-1) list ( -- the ith list in gLists is the set of degree i generators flatten entries mingens ( J^(maxDeg - i) * sub(ideal(gLists#i), ring I) ) ); return trim ideal (flatten gs); )

A method for computing the class [X] A (PN ) of an arbitrary scheme X is needed for our ∈ ∗ applications. The multiplicity of X along an irreducible component Z is multiplicity of X at a general point z Z. This is also the degree of the projectivized tangent cone to z in X, which can ∈ be calculated via the Hilbert polynomial of the graded ring associated to O(X)/I. cycleClass = method() cycleClass ProjectiveScheme := X -> ( if X.CycleClass === null then ( mPrimes := minimalPrimes X.Ideal; -- irreducible components of X X.CycleClass = sum apply (mPrimes, irrComp -> ( hilb := hilbertPolynomial( tangentCone(sub(irrComp, X.CoordinateRing)) ); d := dim hilb; -- dimension of the associated scheme to i m := (hilb#d); -- its geometric multiplicity in X m * (X.Hyperplane)^(dim(X.AmbientSpace)-d)

89 )) ); X.CycleClass )

Some other help functions for a new type can be useful. degree ProjectiveScheme := X -> ( if X.degree === null then ( X.degree = degree(X.Ideal) ); X.degree ) dim ProjectiveScheme := X -> ( if X.dim === null then ( X.dim = dim(variety(X.Ideal)) ); X.dim ) codim(ProjectiveScheme) := {} >> opts -> (X) -> ( if X.codim === null then ( X.codim = dim(X.AmbientSpace) - dim(X) ); X.codim )

-- a "good" hyperplane section H on Y (relative to X) is one -- which does not contain any distinguished varieties of X goodHyperplaneSection := (X,Y) -> ( ds := distinguished ( sub(X.Ideal, Y.CoordinateRing) ); while (true) do ( h := random(1, ring(Y.Ideal)); found := true; -- choose a random hyperplane section of Y -- test to see if it contains any distinguished varieties of X -- if so, start over for d in ds do ( if isSubset(ideal h, d) then ( found = false; break; ) );

90 if found then return h ) ) projDegSaturate := (X,Y) -> ( ideals := for i from 1 to (dim Y) list ( ideal(sum apply(X.Equations, g -> random(0,ring(X.Ideal))*g)) ); hyps := sum(ideals);

if char(X.BaseField) > 0 then ( I := saturate( Y.Ideal + hyps, X.Ideal, Strategy => "F4"); return degree I ) else ( I := saturate( Y.Ideal + hyps, X.Ideal); return degree I ) ) projDegHelmer = method() projDegHelmer(Ideal,Ideal) := (X,Y) -> ( S := ring(X); kk := coefficientRing ring(X); St := kk(monoid[gens S, getSymbol "T"]); SX := sub(X,St); SY := sub(Y,St); varsS := (v -> sub(v,St)) \ (gens S); n := numgens X; Pols := sum(dim variety Y, jj -> ideal sum (n, i -> random(kk)*SX_i)); LA := ideal( sum (numgens S, i -> random(kk)*varsS_i) - 1 ); VS := ideal( sum ( n, i -> 1 - (last gens St)*random(kk)*SX_i) ); Wt := SY + Pols + VS + LA; return numColumns basis cokernel leadTerm gb(Wt); ) projDegHelmer(ProjectiveScheme,ProjectiveScheme) := (X,Y) -> ( projDegHelmer(X.Ideal,Y.Ideal) ) restrictToHplaneSection = method() restrictToHplaneSection(ProjectiveScheme, Thing) := (X,h) -> ( -- here X is the scheme we’re restricting and -- h is a hyperplane (degree 1 element of polynomial ring)

91 N := dim(X.AmbientSpace); kk := X.BaseField; P := kk(monoid[ (i -> (getSymbol "w")_i ) \ (0..N-1) ]); R := ring(X.Ideal); coordchangeIdeal := sub(X.Ideal,{R_N => h}); restrictedIdeal := sub(coordchangeIdeal, {R_N => 0}|((i -> R_i => P_i) \ toList(0..N-1)) ); return restrictedIdeal )

Here is the main method. It can be called on a pair of ideals or ProjectiveSchemes. If called with only one argument, the second is assumed to be the ambientSpace. Start with the header and alternative deﬁnitions. segreClass = method(TypicalValue => RingElement, Options => {Testing => false, Strategy => "Saturate"}) segreClass(Ideal) := opts -> (iX) -> ( iY := trim ideal 0_(ring iX); segreClass(iX, iY, opts) ) segreClass(Ideal,Ideal) := opts -> (iX,iY) -> ( Y := projectiveScheme(iY); X := projectiveScheme(iX, SuperScheme => Y, MakeBaseOfLinearSystem => true); segreClass(X, Y, opts) ) segreClass(ProjectiveScheme) := opts -> (X) -> ( iX := X.Ideal; iY := trim ideal 0_(ring iX); Y := projectiveScheme(iY, AmbientSpace => X.AmbientSpace); segreClass(X, Y, opts) )

Here is the implementation of algorithm 1. segreClass(ProjectiveScheme,ProjectiveScheme) := opts -> (X,Y) -> ( if not ( X.AmbientSpace === Y.AmbientSpace ) then ( error "Expected ProjectiveSchemes with the same AmbientSpace" ); H := X.Hyperplane; if dim X < 0 then (return 0*H); N := dim X; d := first degree ( (X.Ideal)_0 ); -- degree of each generator

92 X0 := X; Y0 := Y;

eqns := while ( dim X >= 0 ) list ( pdeg := if opts.Strategy == "Helmer" then ( projDegHelmer(X,Y) ) else ( projDegSaturate(X,Y) ); D := ( d^(dim Y) * degree(Y) ) - pdeg; D ) do ( hyp := goodHyperplaneSection(X,Y); IY := restrictToHplaneSection(Y,hyp); IX := sub(restrictToHplaneSection(X,hyp), ring(IY)); Y = projectiveScheme IY; X = projectiveScheme(IX, SuperScheme => Y, MakeBaseOfLinearSystem => true); );

matC := sub(segreAlgCoefficientMatrix(dim Y0, dim X0, d), ZZ/32479); vecD := sub(transpose matrix {eqns}, ZZ/32479); vecA := flatten entries solve(matC,vecD);

seg := sum ( for i from 0 to N list ( lift(vecA#i,ZZ) * H^(X0.AmbientSpace.dim - i) ) ); sub(seg,intersectionRing Y0) )

The rest of the code provides functions for computing Chern-Mather classes, Chern-Schwartz- MacPherson classes, polar ranks, and intersection products (in special cases). Here are two helper functions for computing the Chern classes.

RingElement ** AbstractSheaf := (s, L) -> ( c := chern L; R := ring s;

return sum apply( terms s, t -> ( codimn := first degree(t_R); return t * (c^(-codimn)); )) )

93 projectToHypersurface = method() projectToHypersurface(Ideal) := (X) -> ( c := codim X; n := dim variety X; R := ring X; kk := coefficientRing R; L := sum( n+2, i -> ideal(random(1,R)) ); pr := map(R,kk(monoid[(i -> (getSymbol "a")_i )\(1..n+2)]), gens L); return trim (preimage_pr X) ) chernMather = method() chernMather(ProjectiveScheme) := (X) -> ( if codim(X) > 1 then ( << "Projecting to get lower codimension... " << "Currently, codim = " << codim(X) << " in PP^" << dim(X.AmbientSpace) << endl; return chernMather( projectToHypersurface(X.Ideal) ) );

if codim(X) == 1 then (<< "X has codimension 1..." << endl;); cX := cycleClass X; T := tangentBundle(X.AmbientSpace); O := OO_(X.AmbientSpace); iJ := (singularLocus X.Ideal).ideal;

if dim variety iJ < 0 then return chern(T) * cX * (1+cX)^(-1);

s := segreClass(iJ,X.Ideal); a := sub(adams(-1,s), intersectionRing X);

return chern(T) * ( cX * (1+cX)^(-1) + (a ** O(cX) ) ) ) chernMather(Ideal) := (iX) -> ( X := projectiveScheme(iX); return chernMather(X) ) chernSchwartzMacPherson = method() chernSchwartzMacPherson(ProjectiveScheme) := (X) -> ( cX := cycleClass X; T := tangentBundle(X.AmbientSpace);

94 O := OO_(X.AmbientSpace);

iJ := (singularLocus X.Ideal).ideal;

iM := if codim X > 1 then ( ideal ( for i from 1 to codim X - 1 list sum apply(X.Equations, e -> random(0,ring e)*e) ) ) else if codim X == 1 then ( if dim variety iJ < 0 then return chern(T)*cX*(1+cX)^(-1); trim ideal 0_(ring X.Ideal) ) else if codim X == 0 then ( return chern tangentBundle X.AmbientSpace ) else ( error "got codimension of " | toString codim X | " for " | toString X; ) M := projectiveScheme(iM, AmbientSpace => X.AmbientSpace);

J := projectiveScheme(iJ, AmbientSpace => X.AmbientSpace); seg := segreClass(J.Ideal, M.Ideal); s := chern(O(cX)) * sub(seg, intersectionRing X); a := sub(adams(-1,s), intersectionRing X); return chern(T) * ( cX * (1+cX)^(-1) + (a ** O(cX) ) ) ) chernSchwartzMacPherson(Ideal) := (iX) -> ( return chernSchwartzMacPherson(projectiveScheme(iX)) )

Now some additional ways to get at the conormal variety and polar ranks. dualDegree = method() dualDegree(Ideal) := (X) -> ( {* Compute the degree of the dual of a hypersurface via its singularity Segre class *} d := degree X; n := #(gens ring X)-1; J := ideal singularLocus X; seg := segreClass(J, X); A := ring seg; S := sum(n, i -> binomial(n-1,i) * (d-1)^i * seg_((A_1)^(n-i))) return d*(d-1)^(n-1) - S )

95 dualDegree(ProjectiveScheme) := (X) -> ( dualDegree(X.Ideal) ) polarRanks = method() polarRanks(Ideal) := (X’) -> ( X := if codim X’ > 1 then ( projectToHypersurface X’ ) else if codim X’ == 1 then (X’) else error "X should be a proper subvariety"; d := degree X; n := dim variety X; seg := segreClass(ideal singularLocus X, X); A := ring seg; S := sum(k-1, i -> binomial(k,i) * (d-1)^i * seg_((A_1)^(k+1-i)) ); ranks := for k from 0 to n list ( d*(d-1)^(k) - S ); return ranks ) polarRanks(ProjectiveScheme) := (X) -> ( polarRanks(X.Ideal) )

Finally, we give a ﬁrst attempt at an intersectionProduct for arbitrary subvarieties of a projective varieties. Using the techniques from section 2.6.5, we can make a little headway, but we’ll need to compute Segre classes in subvarieties of products of projective space to get a more robust method. The Segre embedding of such a product is far to expensive to compute in, even for very small ambient spaces. intersectionProduct = method() intersectionProduct ( ProjectiveScheme, ProjectiveScheme, ProjectiveScheme ) := (X,V,Y) -> ( {* computes the intersection product X *_Y V corresponding to a fibre square where i is a regular embedding of codimension d (see [F,Chapter 6])

96 j W -----> V | | g| |f | | X -----> Y i *}

-- this error can be avoided if the user just -- passes ideals instead of ProjectiveSchemes if not (V.AmbientSpace === Y.AmbientSpace and X.AmbientSpace === Y.AmbientSpace) then error "expected U,V to be subschemes of Y";

-- check if codim in S.AmbientSpace matches # of equations isCompleteIntersection := S -> ( #(S.Equations) == codim(S) );

H := Y.Hyperplane; -- if X,Y are complete intersections in PP^N, -- we can compute the intersection product easily (a,b,c) := if all({X,Y}, S -> isCompleteIntersection S) then ( -- compute the normal bundle N_X Y chernRoots := S -> apply(S.Equations, eq -> (first degree eq)*H); chernX := product apply(chernRoots X, a -> 1+a); chernY := product apply(chernRoots Y, a -> (1+a)^(-1));

-- compute the Segre class s(W,V) s := segreClass(X,V); (chernX, chernY, s) ) else error "not implemented yet";

-- {a*b*c}_d is the class we want d := dim V - (dim Y - dim X); n := dim Y.AmbientSpace; coefficient(H^(n-d),a*b*c)*H^(n-d) )

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103 BIOGRAPHICAL SKETCH

Corey Harris received his Bachelor of Science in Mathematics from The University of Texas at Austin. After receiving his Ph.D., he will be taking a post-doctoral position at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany.

104