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Tables of Contents for Computations in Algebraic Geometry With Macaulay 2
Chapter/Section Title
Page #
Page Count
Preface
v
List of Contributors
xv
Part I Introducing Macaulay
2
323
Ideals, Varieties and Macaulay 2
3
14
Bernd Sturmfels
A Curve in Affine Three-Space
3
1
Intersecting Our Curve With a Surface
4
2
Changing the Ambient Polynomial Ring
6
2
Monomials Under the Staircase
8
4
Pennies, Nickels, Dimes and Quarters
12
5
References
15
2
Projective Geometry and Homological Algebra
17
24
David Eisenbud
The Twisted Cubic
18
2
The Cotangent Bundle of P3
20
4
The Cotangent Bundle of a Projective Variety
24
2
Intersections by Serre's Method
26
2
A Mystery Variety in P3
28
13
Appendix A. How the ``Mystery Variety'' was Made
37
3
References
40
1
Data Types, Functions, and Programming
41
14
Daniel R. Grayson
Michael E. Stillman
Basic Data Types
41
3
Control Structures
44
2
Input and Output
46
2
Hash Tables
48
4
Methods
52
1
Pointers to the Source Code
53
2
References
53
2
Teaching the Geometry of Schemes
55
18
Gregory G. Smith
Bernd Sturmfels
Distinguished Open Sets
55
1
Irreducibility
56
2
Singular Points
58
2
Fields of Definition
60
1
Multiplicity
61
1
Flat Families
62
1
Bezout's Theorem
63
1
Constructing Blow-ups
64
1
A Classic Blow-up
65
3
Fano Schemes
68
5
References
70
3
Part II Mathematical Computations
Monomial Ideals
73
28
Serkan Hosten
Gregory G. Smith
The Basics of Monomial Ideals
74
3
Primary Decomposition
77
6
Standard Pairs
83
6
Generic Initial Ideals
89
6
The Chain Property
95
6
References
99
2
From Enumerative Geometry to Solving Systems of Polynomial Equations
101
30
Frank Sottile
Introduction
101
2
Solving Systems of Polynomials
103
9
Some Enumerative Geometry
112
2
Schubert Calculus
114
7
The 12 Lines: Reprise
121
10
References
128
3
Resolutions and Cohomology over Complete Intersections
131
48
Luchezar L. Avramov
Daniel R. Grayson
Matrix Factorizations
133
6
Graded Algebras
139
2
Universal Homotopies
141
4
Cohomology Operators
145
5
Computation of Ext Modules
150
7
Invariants of Modules
157
13
Invariants of Pairs of Modules
170
9
Appendix A. Gradings
176
1
References
177
2
Algorithms for the Toric Hilbert Scheme
179
36
Michael Stillman
Bernd Sturmfels
Rekha Thomas
Generating Monomial Ideals
182
6
Polyhedral Geometry
188
5
Local Equations
193
6
The Coherent Component of the Toric Hilbert Scheme
199
16
Fourier-Motzkin Elimination
206
5
Minimal Presentation of Rings
211
2
References
213
2
Sheaf Algorithms Using the Exterior Algebra
215
36
Wolfram Decker
David Eisenbud
Introduction
215
3
Basics of the Bernstein-Gel'fand-Gel'fand Correspondence
218
4
The Cohomology and the Tate Resolution of a Sheaf
222
4
Cohomology and Vector Bundles
226
4
Cohomology and Monads
230
6
The Beilinson Monad
236
5
Examples
241
10
References
247
4
Needles in a Haystack: Special Varieties via Small Fields
251
30
Frank-Olaf Schreyer
Fabio Tonoli
How to Make Random Curves up to Genus 14
253
10
Comparing Green's Conjecture for Curves and Points
263
4
Pfaffian Calabi-Yau Threefolds in P6
267
14
References
277
4
D-modules and Cohomology of Varieties
281
44
Uli Walther
Introduction
282
3
The Weyl Algebra and Grobner Bases
285
7
Bernstein-Sato Polynomials and Localization
292
12
Local Cohomology Computations
304
9
Implementation, Examples, Questions
313
12
References
321
4
Index
325