Computations in algebraic geometry with macaulay 2代数几何与Macaulay2的计算
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作者： David Eisenbud，Daniel R. Grayson等著
出 版 社： 湖南文艺出版社
出版时间： 2001-10-1字数：版次：页数： 329印刷时间： 2001/10/01开本： 16开印次：纸张： 胶版纸I S B N ： 9783540422303包装： 精装内容简介
This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.
目录
Preface
List of Contributors
Part I Introducing Macaulay 2
Ideals, Varieties and Macaulay 2
Bernd Sturmfels
1A Curve in Affine Three-Space
2Intersecting Our Curve With a Surface
3Changing the Ambient Polynomial Ring
4Monomials Under the Staircase
5Pennies, Nickels, Dimes and Quarters
References
Projective Geometry and Homological Algebra
David Eisenbud
1The Twisted Cubic
2The Cotangent Bundle of I?3
3The Cotangent Bundle of a Projective Variety
4Intersections by Serre's Method
5A Mystery Variety in ]?3
Appendix A. How the "Mystery Variety" was Made
References
Data Types, Fhlnctions, and Programming
Daniel R. Grayson and Michael E. Stillman
1Basic Data Types
2Control Structures
3Input and Output
4Hash Tables
5Methods
6Pointers to the Source Code
References
Teaching the Geometry of Schemes
Gregory G. Smith and Bernd Sturmfels
1Distinguished Open Sets
2Irreducibility
3Singular Point
4Fields of Definition
5Multiplicity
6Flat Families
7Bezout's Theorem
8Constructing Blow-ups
9A Classic Blow-up
10 Fano Schemes
References
Part II Mathematical Computations
Monomial Ideals
Serkan Hosaten and Gregory G. Smith
1The Basics of Monomial Ideals
2Primary Decomposition
3Standard Pairs
4Generic Initial Ideals
5The Chain Property
References
From Enumerative Geometry to Solving Systems of Polynomial Equations
Frank Sottile
1Introduction
2Solving Systems of Polynomials
3Some Enumerative Geometry
4Schubert Calculus
5The 12 Lines: Reprise
References
Resolutions and Cohomology over Complete Intersections
Luchezar L. Avramov and Daniel R. Grayson
1Matrix Factorizations
2Graded Algebras
3UniversM Homotopies
4Cohomology Operators
5Computation of Ext Modules
6Invariants of Modules
7Invariants of Pairs of Modules
Appendix A. Gradings
References
Algorithms for the Toric Hilbert Scheme
Sheaf Algorithms Using the Exterior Algebra
Needles in a Haystack:Special Varieties via Small Fields
D-modules and Cohomology of Varieties
Index