Combinatorial Aspects of Commutative Algebra and Algebraic Geometry: The Abel Symposium 2009 (Abel Symposia Book 6)

The Abel Symposium 2009 "Combinatorial aspects of Commutative Algebra and Algebraic Geometry", held at Voss, Norway, featured talks by leading researchers in the field. This is the proceedings of the Symposium, presenting contributions on syzygies, tropical geometry, Boij-Söderberg theory, Schubert calculus, and quiver varieties. The volume also includes an introductory survey on binomial ideals with applications to hypergeometric series, combinatorial games and chemical reactions. The contributions pose interesting problems, and offer up-to-date research on some of the most active fields of commutative algebra and algebraic geometry with a combinatorial flavour. Preface to the Series 6 Preface 7 Contents 9 Contributors 13 The Cone of Betti Diagrams of Bigraded Artinian Modules of Codimension Two 15 1 Introduction 15 2 Preliminaries 17 2.1 Betti Diagrams and the Multigraded Herzog--Kühl Equations 17 2.2 The Equivariant Resolution 19 2.3 The Linear Space of Betti Diagrams of Multigraded Artinian Modules 19 2.4 The Linear Space in the Case of Two Variables 20 3 The Positive Cone of Bigraded Betti Diagrams 21 3.1 Partitions 21 3.2 Decomposing 23 4 Existence of Resolutions 25 5 Resolutions of Trigraded Artinian Modules of Codimension Three 28 References 30 Koszul Cycles 31 1 Introduction 31 2 Notation and Generalities 32 3 Bounds for Koszul Cycles 35 4 Green--Lazarsfeld Index for Segre--Veronese Rings 40 5 Generating Koszul Cycles 43 References 46 Boij--Söderberg Theory 48 1 Introduction 48 2 Betti Tables 49 3 Facets of the Cone and Cohomology Tables 52 4 Sheaves on a Subvariety 55 5 Several Line Bundles 56 References 60 Powers of Componentwise Linear Ideals 62 1 Introduction 62 2 The Criterion 64 3 Chordal Graphs 68 References 73 Modules with 1-Dimensional Socle and Components of Lusztig Quiver Varieties in Type A 74 1 Introduction 74 1.1 Acknowledgements 75 2 Background 75 2.1 Notation 75 2.2 The Preprojective Algebra 76 2.3 Socle of Modules 76 2.4 Lusztig Quiver Varieties 77 3 Modules with One-Dimensional Socle 77 3.1 The Maya Modules 77 3.2 The Uniqueness Theorem 78 3.3 Computation of Hom Spaces 81 3.4 Connection with MV Polytopes 82 4 Description of Irreducible Components 83 4.1 Savage's Description of the Components 83 4.2 Description of the Components by Hom Spaces 84 References 85 Realization Spaces for Tropical Fans 86 1 Introduction 86 2 Realization and Compactification 89 3 Representability 91 4 Tropical Realization of Matroid Fans 92 5 Murphy's Law for Tropical Realization Spaces 96 References 100 A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus 102 1 Introduction 102 2 Partitions 103 3 Factorial Schur Polynomials 103 4 Factorial Schur Polynomials over Polynomial Rings 106 5 Factorial Schur Polynomials and Schubert Classes 107 References 110 Theory and Applications of Lattice Point Methods for Binomial Ideals 111 1 Introduction 111 Theory 112 2 Affine Semigroups and Prime Binomial Ideals 112 2.1 Affine Semigroups 112 2.2 Affine Semigroup Rings 116 2.3 Prime Binomial Ideals 118 3 Monomial Ideals and Primary Binomial Ideals 120 3.1 Monomial Primary Ideals 120 3.2 Congruences on Monoids 123 3.3 Binomial Primary Ideals with Monomial Associated Primes 125 4 Binomial Primary Decomposition 128 4.1 Monomial Primes Minimal over Binomial Ideals 128 4.2 Primary Components for Arbitrary Given Associated Primes 131 4.3 Finding Associated Primes Combinatorially 133 Applications 137 5 Hypergeometric Series 137 5.1 Binomial Horn Systems 137 5.2 True Degrees and Quasidegrees of Graded Modules 140 5.3 Counting Series Solutions 143 6 Combinatorial Games 146 6.1 Introduction to Combinatorial Game Theory 146 6.2 Lattice Games 150 6.3 Rational Strategies 153 6.4 Misère Quotients 156 7 Mass-Action Kinetics in Chemistry 158 7.1 Binomials from Chemical Reactions 159 7.2 Global Attractor Conjecture 161 References 163 Equations Defining Secant Varieties: Geometry and Computation 167 1 Introduction 167 2 Computing Secant Varieties 170 2.1 Secant Varieties via Elimination 171 2.2 Secant Varieties via Prolongation 172 2.3 Compute the Ideal of a Smooth Curve 173 3 Secant Varieties as Vector Bundles: Terracini Recursion 176 3.1 Secant Varieties of Points 177 3.2 Blowing up a Rational Normal Curve of Degree 3 178 3.3 Blowing up a Rational Normal Curve of Degree 4 179 3.4 Cohomology Along the Fibers 181 Appendix A Code for Computing Prolongations 183 References 185 The Abel Symposium 2009 'Combinatorial aspects of Commutative Algebra and Algebraic Geometry', held at Voss, Norway, featured talks by leading researchers in the field. This is the proceedings of the Symposium, presenting contributions on syzygies, tropical geometry, Boij-Soderberg theory, Schubert calculus, and quiver varieties. The volume also includes an introductory survey on binomial ideals with applications to hypergeometric series, combinatorial games and chemical reactions. The contributions pose interesting problems, and offer up-to-date research on some of the most active fields of c The Abel Symposium 2009 "Combinatorial aspects of Commutative Algebra and Algebraic Geometry", held at Voss, Norway, featured talks by leading researchers in the field. This is the proceedings of the Symposium, presenting contributions on syzygies, tropical geometry, Boij-Soederberg theory, Schubert calculus, and quiver varieties.