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Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry Corey S Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2017 Effective Methods in Intersection Theory and Combinatorial Algebraic Geometry Corey S. (Corey Scott) Harris 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 & SCIENCES EFFECTIVE METHODS IN INTERSECTION THEORY AND COMBINATORIAL ALGEBRAIC GEOMETRY By 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 Copyright c 2017 Corey S. Harris. All Rights Reserved. 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
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