Western University Scholarship@Western Electronic Thesis and Dissertation Repository 6-25-2015 12:00 AM Algorithms to Compute Characteristic Classes Martin Helmer The University of Western Ontario Supervisor Eric Schost The University of Western Ontario Graduate Program in Applied Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Martin Helmer 2015 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Algebra Commons, Algebraic Geometry Commons, Geometry and Topology Commons, Other Applied Mathematics Commons, and the Theory and Algorithms Commons Recommended Citation Helmer, Martin, "Algorithms to Compute Characteristic Classes" (2015). Electronic Thesis and Dissertation Repository. 2923. https://ir.lib.uwo.ca/etd/2923 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Algorithms to Compute Characteristic Classes (Thesis Format: Integrated-Article) by Martin Helmer Graduate Program in Applied Mathematics A thesis submitted in partial fulfilment of the requirements for a Doctor of Philosophy The School of Graduate and Postdoctoral Studies The University of Western Ontario London, Ontario, Canada c Martin Helmer 2015 i Abstract In this thesis we develop several new algorithms to compute characteristic classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (cSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases). We begin with subschemes of a projective space Pn over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the cSM class, Segre class and the Euler characteristic. The algorithms can be im- plemented using either symbolic or numeric methods. The algorithms are based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Running time bounds are given for these algorithms and the al- gorithms are found to perform favourably compared to other applicable algorithms. Relations between our algorithms and other existing algorithms are explored. In the special case of a complete intersection subcheme we develop a second algo- rithm to compute cSM classes and Euler characteristics in a more direct and efficient manner. Each of these algorithms are generalized to subschemes of Pn1 × · · · × Pnm . Running time bounds for the generalized algorithms to compute the cSM class, Segre class and the Euler characteristic are given. Our Segre class algorithm is tested in com- parison to another applicable algorithm and is found to perform favourably. To the best of our knowledge there are no other algorithms in the literature which compute the cSM class and Euler characteristic in the multi-projective setting. For complete simplical toric varieties defined by a fan we give a strictly combi- natorial algorithm to compute the cSM class and Euler characteristic and a second combinatorial algorithm with reduced running time to compute only the Euler char- acteristic. We also prove several Bezout´ type bounds in multi-projective space. An application ii of these bounds to obtain a sharper degree bound on a certain system with a natural bi-projective structure is demonstrated. Keywords: Euler characteristic; Chern-Schwartz-MacPherson class; Segre class; Characteristic class; Computer algebra; Computational intersection theory; Alge- braic geometry iii Acknowledgements The author of this thesis would like thank Michael Stillman for suggesting to us the problem considered in Chapter5 of this thesis; that is the problem of computing Chern-Schwartz-MacPherson classes of toric varieties. The author would also like to thank Eric´ Schost for numerous helpful comments and discussions regarding the content of this thesis. iv Contents Abstracti Acknowledgements iii List of Tables vii List of Appendices viii 1 Introduction1 1.1 Overview of Contributions ...................... 2 1.1.1 Computing Characteristics Classes.............. 2 1.1.2 Bezout-like´ Results in Multi-projective Space........ 7 1.2 Review and Previous Work...................... 8 1.2.1 The Setting .......................... 8 1.2.2 The Segre Class........................ 11 1.2.3 The Chern-Schwartz-MacPherson Class and the Euler Char- acteristic............................ 12 1.3 Results................................. 15 1.3.1 Characteristic Class Computations in Pn . 15 1.3.2 Characteristics Class Computations in Pn1 × · · · × Pnm . 24 1.3.3 Computing the Chern-Schwartz-MacPherson Class and Eu- ler Characteristic of Complete Simplicial Toric Varieties . 28 1.3.4 Bezout´ Type Results in Multi-projective Space . 31 2 Computing Characteristics Classes in Projective Space 38 2.1 Problem and Setting.......................... 41 2.1.1 Chow Groups and Chow Rings................ 41 2.1.2 Chern and Chern-Schwartz-MacPherson Classes . 45 2.2 Review................................. 48 2.2.1 Projective Degrees ...................... 49 2.2.2 Segre classes ......................... 50 v 2.2.3 Chern-Schwartz-MacPherson Classes ............ 54 2.3 Main Results and Algorithms..................... 55 2.3.1 Results ............................ 56 2.3.2 Algorithms .......................... 61 2.4 Performance.............................. 64 2.4.1 Timings for the Compution of Segre Classes......... 67 2.4.2 Timings for the Compution of cSM Classes and Euler Char- acteristics........................... 68 2.4.3 Running Time Bounds .................... 70 3 An Improved Algorithm for Complete Intersections 77 3.1 Background.............................. 79 3.2 Main Results and Algorithms..................... 82 3.2.1 The Main Result ....................... 83 3.2.2 Algorithms .......................... 87 3.3 Performence.............................. 89 4 Algorithms to Compute the Topological Euler Characteristic and the Chern-Schwartz-MacPherson Class of Arbitrary Subschemes of Prod- ucts of Projective Space 97 4.1 Review................................. 99 4.1.1 Pn1 × · · · × Pnm and its Chow Ring.............. 99 4.1.2 Previous Segre Class Algorithms in Pn1 × · · · × Pnm . 101 4.1.3 cSM Class of a Hypersurface For a Subscheme of any Smooth Variety.............................102 4.2 Main Results .............................102 4.2.1 The Segre Class of Subvarieties of Pn1 × · · · × Pnm . 103 4.2.2 Computing the Projective Multi-degrees . 105 4.2.3 The cSM Class of Complete Intersections . 110 4.3 Algorithms for Subschemes of a Product of Projective Spaces . 111 4.3.1 Segre Class ..........................112 4.3.2 The cSM Class Via Inclusion/Exclusion . 114 4.3.3 cSM: The Complete Intersection Case . 115 4.4 Performance..............................116 4.4.1 Run Time Tests........................116 4.4.2 Running Time Bounds ....................120 5 cSM Classes of Complete Simplicial Toric Varieties 124 5.1 Review.................................125 5.1.1 Constructing the Toric Variety of a Fan . 125 vi 5.1.2 Orbits and Orbit Closures...................130 5.1.3 Complete Simplicial Toric Varieties . 131 5.2 The Chow Ring of a Complete Simplicial Toric Variety . 132 5.3 Algorithms to Compute the Chern-Schwartz-MacPherson Class and Euler Characteristic of a Complete Simplicial Toric Variety . 134 5.4 Performance..............................141 6 Bezout´ Type results in Multi-Projective Space For Application to Au- tomated Polynomial System Solving 145 6.1 Review.................................147 6.2 Bezout-like´ Results..........................151 6.3 Applications..............................153 6.3.1 Relations Between the Geometric Multiplicity and Other Multiplicity Functions ....................154 6.3.2 Affine Varieties with a Bi-projective Structure . 158 7 Conclusion 165 A Overview of Implementations and Lists of Examples 170 A.1 Overview of the Implementation used in Chapter2 and Chapter3 . 170 A.2 Examples From Chapter2.......................172 A.3 Examples From Chapter3 ......................174 A.4 Overview of the Implementation used in Chapter4 . 177 A.5 Examples From Chapter4.......................179 A.6 Overview of the Implementation used in Chapter5 . 182 Curriculum Vitae 185 vii List of Tables 2.1 Comparision of algorithms to compute the Segre class of a sub- scheme of Pn .............................. 67 2.2 Comparison of algorithms to compute the cSM class of a subscheme of Pn using inclusion/exclusion.................... 70 3.1 Run time results for Algorithm 3.2.2 working over Q . 90 3.2 Run time results for Algorithm 3.2.2 working over the finite field GF(32749)............................... 93 4.1 Comparision of algorithms to compute the Segre class of a sub- scheme of Pn1 × · · · × Pnm . 118 4.2 The cSM class and Euler characteristic in multi-projective space . 119 4.3 The cSM class and Euler characteristic of certain compmplete inter- sections in multi-projective space...................119 5.1 The cSM class and Euler chacteristic of compete simplicial toric va- rieties .................................143 viii List of Appendices AppendixA...................................170 1 Chapter 1 Introduction