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Secant Indices, Duality Defect, and Generalizations of the Segre Zeta

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Secant Indices, Duality Defect, and Generalizations of the Segre Zeta Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School Secant Indices, Duality Defect, and Graeysnone S.r Joargleinzsoantions of the Segre Zeta Function Follow this and additional works at the DigiNole: FSU's Digital Repository. For more information, please contact [email protected] FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES SECANT INDICES, DUALITY DEFECT, AND GENERALIZATIONS OF THE SEGRE ZETA FUNCTION By GRAYSON S. JORGENSON A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2020 Copyright c 2020 Grayson S. Jorgenson. All Rights Reserved. Grayson S. Jorgenson defended this dissertation on March 30, 2020. The members of the supervisory committee were: Paolo Aluffi Professor Directing Dissertation Svetlana Pevnitskaya University Representative Ettore Aldrovandi Committee Member Mark van Hoeij Committee Member Kathleen Petersen Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the dissertation has been approved in accordance with university requirements. ii ACKNOWLEDGMENTS I wish to thank Paolo Aluffi for all his guidance, encouragement, and support over the duration of my doctoral work. I always look forward to meeting with him and I have greatly enjoyed hearing his insights. His passion for math and for science and the arts as a whole has been an enormous inspiration to me, helping to drive my own work and curiosity. I am extremely grateful to have been able to work under his tutelage. Throughout my degree I felt fortunate to have been able to get to know many of the other great people in the FSU math department. In particular, I would like to thank the professors who taught me in a variety of courses in the earlier stages of my studies: Ettore Aldrovandi, Phil Bowers, Sergio Fenley, Wolfgang Heil, Alec Kercheval, Eric Klassen, Craig Nolder, and Mark van Hoeij. Thanks also to my fellow grad students, especially Arun, Ben, Braulio, Michael, Minfa, and Yi for many fun times and discussions. I would also like to express here my gratitude to Benjamin Hutz to whom I owe a large debt for playing a pivotal role in introducing me to the world of pure math during my undergraduate studies and for inviting me into several great opportunities over the years. Finally, I wish to thank my parents and sister for all of their love and support. iii TABLE OF CONTENTS List of Figures . vi Abstract . vii 1 Introduction 1 1.1 Secant indices . .1 1.2 Duality defect . .2 1.3 Relative Segre zeta functions . .5 1.4 Segre Zeta functions in the equivariant setting . .6 2 Secant Indices 9 2.1 Introduction . .9 2.2 General properties . 10 2.2.1 Strictness of growth . 12 2.3 Questions and a guiding principle . 16 2.4 Veronese varieties . 23 2.5 Segre varieties . 36 2.6 Lines on surfaces . 41 3 Linear Recurrence Sequences and the Duality Defect Conjecture 45 3.1 Introduction . 45 3.2 Preliminaries . 48 3.3 Dual varieties and duality defect . 49 3.4 A generalized algorithm and bounds for the degrees of counterexamples . 54 3.5 Duality defect in codimension 3 . 60 4 A Relative Segre Zeta Function 64 4.1 Introduction . 64 4.1.1 Motivation . 64 4.1.2 Statement of the results . 66 4.2 Proof of the main result . 70 4.3 Extension of bundles and sections . 75 4.4 The case of subbundles . 77 4.4.1 When F = E ⊕ L .................................. 77 4.4.2 Reduction of the ambient space dimension . 79 4.5 Examples . 80 4.5.1 Subschemes of projective space . 80 4.5.2 Subschemes of products of projective spaces . 81 5 Equivariant Realization of the Segre Zeta Function 84 5.1 Introduction . 84 5.2 Segre classes . 86 iv 6 Conclusion 90 Bibliography . 92 Biographical Sketch . 96 v LIST OF FIGURES (2) 2 2.1 Successive hyperplane sections of v3 (P )......................... 25 2.2 Successive monomial additions to J for n = 2, d =4................... 26 vi ABSTRACT This thesis consists of three independent projects in the field of algebraic geometry. The first of these is the focus of Chapter 2. There we define a family of integer sequences we n refer to as the family of secant indices of projective subvarieties. If X ⊆ P is a subvariety, its sequence of secant indices is denoted L(X) and records the maximal finite intersections of X with n linear subvarieties of P . We prove several properties about these sequences, develop a method to compute term-wise lower bounds and compute these lower bounds for Veronese and Segre varieties. We state several questions and conjectures regarding these sequences which to our knowledge are open. Chapter 3 concerns a project studying the so-called duality defect of projective subvarieties. n Given a projective subvariety X ⊆ P , one may consider its dual subvariety in dual projective space n _ (P ) . For low codimension smooth X, it would be a consequence of Hartshorne's conjecture that the dual of X is always a hypersurface. This expectation is known as the duality defect conjecture, and in Chapter 3 we push a combinatorial approach of Holme and Oaland further to verify that the conjecture holds in the codimension 3 case for projective spaces of odd dimension, and also to derive a degree bound that any counterexamples would have to satisfy. Chapters 4 and 5 concern Segre classes of closed subschemes of projective bundles. By extending a given homogeneous ideal to progressively larger polynomial rings, one obtains an infinite sequence of closed subschemes of progressively higher dimensional projective spaces which are cones over the preceding subschemes. The Segre classes of these subschemes are related by the Segre zeta function defined by Aluffi. In Chapter 4, we generalize this Segre zeta function to subschemes of projective bundles over a smooth base variety. In Chapter 5 we explore a realization of the Segre zeta function in the context of equivariant algebraic geometry. vii CHAPTER 1 INTRODUCTION Here we summarize the content of the subsequent chapters while introducing some of the necessary prerequisite material. First, some conventions we will fix throughout. We will take our base field to be k = C, though all the results of Chapter 4 hold when k is any algebraically closed field of characteristic zero. All schemes are assumed to be of finite type over k. A variety for us will always refer to an integral scheme, so in particular is assumed to be irreducible. All points are closed points. 1.1 Secant indices n A fundamental observation in basic algebraic geometry is that if one has a subvariety X of P , n then for any hyperplane H ⊆ P , the dimension of X \ H is at least dim(X) − 1. Furthermore, it is always possible to find hyperplanes so that dim(X \ H) = dim(X) − 1 [32, Section I.7]. A n consequence of this is that if one has a linear subvariety L ⊆ P of dimension r, then dim(X \ L) ≥ dim(X) − n + r. Thus, if n − r ≤ dim(X), then the intersection X \ L must be nonempty. The case where dim(X) = n − r is special in that the smallest possible intersection X \ L will be a finite set of points. For \most" such L, X \ L will be finite. More precisely, the set of linear subvarieties of dimension r is parameterized by the Grassmannian G(r; n). When n − r = dim(X) there is a dense open subset of G(r; n) of linear subvarieties that all meet X in the same finite number of points. This maximal number of points that can appear in such an intersection is known as the degree deg(X) of X. By the same token, when dim(X) < n − r, then most L do not meet X at all; there is a dense open subset of G(r; n) of linear subvarieties with empty intersection with X. Despite this, nonempty intersections can still easily occur. A fundamental question is then to ask about the maximal possible number of points that can occur in such an intersection. Because most linear subvarieties of these smaller dimensions do not meet X, traditional intersection theory techniques 1 for addressing enumerative geometry problems cannot be used to produce exact answers. This question is an instance of a so-called quasi-enumerative geometry problem [20]. n In Chapter 2 we study this problem of special intersections. For a variety X ⊆ P and an integer 0 ≤ i ≤ codim(X) we define an integer Li(X) := maxfjX \ Ljg where this maximum is taken over all dimension i linear subvarieties L with X \ L finite and reduced. We call the collection of these integers as a sequence L(X) := (L0(X);:::; Lcodim(X)(X)) the sequence of secant indices of X. We will derive several properties that these sequences satisfy. In particular, the sequence is always nondecreasing from left to right and the first and last numbers of the sequence are 1 and n deg(X), respectively. When X is not a minimal degree subvariety of P , that is, when deg(X) > codim(X) + 1, there must be at least one gap of size ≥ 2 in the sequence. Because of this, the sequences of secant indices for higher degree varieties seem to possess interesting combinatorial properties. We will investigate the cases where X is a Veronese or Segre variety in detail, obtaining term-wise lower bounds for L(X) that are conjecturally in fact equal to L(X). In the case of Veronese embeddings, the secant indices are related to the Cayley-Bacharach theorem and the open Eisenbud-Green-Harris conjecture.
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