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Table of contents: "Computing in Algebraic Geometry - A Quick Start using Singular"

Preface (ps-file)
0 Introductory Remarks on Computer Algebra
1 Basic Notations and Ideas: A Historical Account
2 Basic Computational Problems and Their Solution
2.1 The Geometry-Algebra Dictionary
2.2 Basic Applications of Gröbner Bases
3. An Introduction to Singular
3.1 General Remarks on Singular and its Syntax
3.2 Rings in Singular
3.2.1 Global Monomial Orders
3.2.2 Creating Ring Maps
3.3 Ideals, Vectors and Modules in Singular
3.4 Handling Graded Modules
3.5 Computing Gröbner Bases
3.6 Basic Applications of Gröbner Bases (revisited)
3.6.1 Ideal Membership Test
3.6.2 Elimination
3.6.3 Kernel of a Ring Map
3.6.4 Test for Subalgebra Membership
3.6.5 Test for Surjectivity of a Ring Map
3.6.6 Syzygies and Free Resolutions
3.7 Gröbner Bases over Noncommutative Algebras
3.8 Writing Singular Procedures and Libraries
3.9 Communication with Other Systems
3.10 Visualization: Plotting Curves and Surfaces
Practical Session I
Practical Session II
4 Homological Algebra I
4.1 Lifting Homomorphisms
4.2 Constructive Module Theory
4.2.1 Cokernels and Mapping Cones
4.2.2 Modulo
4.2.3 Kernel, Hom, Ext, Tor, and more
4.2.4 Some Explicit Constructions
5 Homological Algebra II
5.1 Flatness
5.2 Depth and Codimension
5.3 Cohen-Macaulay Rings
Practical Session III
6 Solving Systems of Polynomial Equations
6.1 Gröbner Basis Techniques
6.1.1 Computing Dimension
6.1.2 Zero-Dimensional Solving by Elimination
6.1.3 Decomposition (Factorizing Buchberger Algorithm, Triangular Decompositions)
6.2 Resultant Based Methods
6.2.1 The Sylvester Resultant
6.2.2 Multipolynomial Resultants
6.2.3 Zero-Dimensional Solving via Resultants
7 Primary Decomposition and Normalization
7.1 Primary Decomposition
7.2 Normalization
Practical Session IV
8 Algorithms for Invariant Theory
8.1 Finite Groups
8.1.1 The Nonmodular Case
8.1.2 The Modular Case
8.1.3 Quotients for Finite Group Actions
8.2 Linearly Reductive Groups
9 Computing in Local Rings
9.1 Rings Implemented by Monomial Orders
9.2 Standard Bases and their Computation
9.3 Factorization and Primary Decomposition
9.4 Computing Dimension
9.5 Elimination
9.6 Hamburger-Noether Expansion
Practical Session V
Appendix A. Sheaf Cohomology and Beilinson Monads
Appendix B. Solutions to Exercises