Computations in Algebraic Geometry with Macaulay 2

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. Table of Contents Cover Computations in algebraic geometry with Macaulay 2 ISBN 3540422307 Preface Table of Contents List of Contributors Part I Introducing Macaulay 2 Ideals, Varieties and Macaulay 2 1 A Curve in A�ne Three-Space 2 Intersecting Our Curve With a Surface 3 Changing the Ambient Polynomial Ring 4 Monomials Under the Staircase 5 Pennies, Nickels, Dimes and Quarters Projective Geometry and Homological Algebra 1 The Twisted Cubic 2 The Cotangent Bundle of P3 3 The Cotangent Bundle of a Projective Variety 4 Intersections by Serre's Method 5 A Mystery Variety in P3 Data Types, Functions, and Programming 1 Basic Data Types 2 Control Structures 3 Input and Output 4 Hash Tables 5 Methods 6 Pointers to the Source Code Teaching the Geometry of Schemes 1 Distinguished Open Sets 2 Irreducibility 3 Singular Points 4 Fields of De nition 5 Multiplicity 6 Flat Families 7 B�ezout's Theorem 8 Constructing Blow-ups 9 A Classic Blow-up 10 Fano Schemes Part II Mathematical Computations Monomial Ideals 1 The Basics of Monomial Ideals 2 Primary Decomposition 3 Standard Pairs 4 Generic Initial Ideals 5 The Chain Property From Enumerative Geometry to Solving Systems of Polynomial Equations 1 Introduction 2 Solving Systems of Polynomials 3 Some Enumerative Geometry 4 Schubert Calculus 5 The 12 Lines: Reprise Resolutions and Cohomology over Complete Intersections 1 Matrix Factorizations 2 Graded Algebras 3 Universal Homotopies 4 Cohomology Operators 5 Computation of Ext Modules 6 Invariants of Modules 7 Invariants of Pairs of Modules Algorithms for the Toric Hilbert Scheme 1 Generating Monomial Ideals 2 Polyhedral Geometry 3 Local Equations 4 The Coherent Component of the Toric Hilbert Sheaf Algorithms Using the Exterior Algebra 1 Introduction 2 Basics of the Bernstein-Gel'fand-Gel'fand 3 The Cohomology and the Tate Resolution of a Sheaf 4 Cohomology and Vector Bundles 5 Cohomology and Monads 6 The Beilinson Monad 7 Examples Needles in a Haystack: Special Varieties via Small Fields 1 How to Make Random Curves up to Genus 14 2 Comparing Green's Conjecture for Curves 3 Pfa�an Calabi-Yau Threefolds in P6 D-modules and Cohomology of Varieties 1 Introduction 2 The Weyl Algebra and Gr�obner Bases 3 Bernstein-Sato Polynomials and Localization 4 Local Cohomology Computations 5 Implementation, Examples, Questions Index