This book presents algorithmic tools for algebraic geometry, with experimental applications. It also introduces Macaulay 2, a computer algebra system supporting research in algebraic geometry, commutative algebra, and their applications. The algorithmic tools presented here are designed to serve readers wishing to bring such tools to bear on their own problems. The first part of the book covers Macaulay 2 using concrete applications; the second emphasizes details of the mathematics.
Author(s): Bernd Sturmfels, David Eisenbud, Daniel R. Grayson, Mike Stillman (Editors)
Edition: 1
Publisher: Springer
Year: 2002
Language: English
Pages: 346
Preface......Page 5
Table of Contents......Page 11
List of Contributors......Page 15
Part I: Introducing Macaulay 2......Page 17
1 A Curve in Affine Three-Space......Page 19
2 Intersecting Our Curve With a Surface......Page 20
3 Changing the Ambient Polynomial Ring......Page 22
4 Monomials Under the Staircase......Page 24
5 Pennies, Nickels, Dimes and Quarters......Page 28
References......Page 31
Projective Geometry and Homological Algebra......Page 33
1 The Twisted Cubic......Page 34
2 The Cotangent Bundle of P3......Page 36
3 The Cotangent Bundle of a Projective Variety......Page 40
4 Intersections by Serre's Method......Page 42
5 A Mystery Variety in P3......Page 44
Appendix A. How the "Mystery Variety" was Made......Page 53
References......Page 56
1 Basic Data Types......Page 57
2 Control Structures......Page 60
3 Input and Output......Page 62
4 Hash Tables......Page 64
5 Methods......Page 68
References......Page 69
1 Distinguished Open Sets......Page 71
2 Irreducibility......Page 72
3 Singular Points......Page 74
4 Fields of Definition......Page 76
5 Multiplicity......Page 77
6 Flat Families......Page 78
7 Bézout's Theorem......Page 79
8 Constructing Blow-ups......Page 80
9 A Classic Blow-up......Page 81
10 Fano Schemes......Page 84
References......Page 86
Part II Mathematical Computations......Page 87
Monomial Ideals......Page 89
1 The Basics of Monomial Ideals......Page 90
2 Primary Decomposition......Page 93
3 Standard Pairs......Page 99
4 Generic Initial Ideals......Page 105
5 The Chain Property......Page 111
References......Page 115
1 Introduction......Page 117
2 Solving Systems of Polynomials......Page 119
3 Some Enumerative Geometry......Page 128
4 Schubert Calculus......Page 130
5 The 12 Lines: Reprise......Page 137
References......Page 144
Resolutions and Cohomology over Complete Intersections......Page 147
1 Matrix Factorizations......Page 149
2 Graded Algebras......Page 155
3 Universal Homotopies......Page 157
4 Cohomology Operators......Page 161
5 Computation of Ext Modules......Page 166
6 Invariants of Modules......Page 173
7 Invariants of Pairs of Modules......Page 186
Appendix A. Gradings......Page 192
References......Page 193
Algorithms for the Toric Hilbert Scheme......Page 195
1 Generating Monomial Ideals......Page 198
2 Polyhedral Geometry......Page 204
3 Local Equations......Page 209
4 The Coherent Component of the Toric Hilbert Scheme......Page 215
Appendix A. Fourier-Motzkin Elimination......Page 222
Appendix B. Minimal Presentation of Rings......Page 227
References......Page 229
1 Introduction......Page 231
2 Basics of the Bernstein-Gel'fand-Gel'fand Correspondence......Page 234
3 The Cohomology and the Tate Resolution of a Sheaf......Page 238
4 Cohomology and Vector Bundles......Page 242
5 Cohomology and Monads......Page 246
6 The Beilinson Monad......Page 252
7 Examples......Page 257
References......Page 263
Needles in a Haystack: Special Varieties via Small Fields......Page 267
1 How to Make Random Curves up to Genus 14......Page 269
2 Comparing Green's Conjecture for Curves and Points......Page 279
3 Pfaffian Calabi-Yau Threefolds in P6......Page 283
References......Page 293
D-modules and Cohomology of Varieties......Page 297
1 Introduction......Page 298
2 The Weyl Algebra and Gröbner Bases......Page 301
3 Bernstein-Sato Polynomials and Localization......Page 308
4 Local Cohomology Computations......Page 320
5 Implementation, Examples, Questions......Page 329
References......Page 337
Index......Page 341
