Gröbner Bases and Algebraic Geometry

Summary

Preface

After the notion of Gröbner bases and an algorithm for constructing them was introduced by Buchberger [Bul, Bu2] algebraic geometers have used Gröbner bases as the main computational tool for many years, either to prove a theorem or to disprove a conjecture or just to experiment with examples in order to obtain a feeling about the structure of an algebraic variety. Nontrivial problems coming either from logic, mathematics or applications usually lead to nontrivial Gröbner basis computations, which is the reason why several improvements have been provided by many people and have been implemented in general purpose systems like Axiom, Maple, Mathematica, Reduce, etc., and systems specialized for use in algebraic geometry and commutative algebra like CoCoA, Macaulay and Singular.

The present paper starts with an introduction to some concepts of algebraic geometry which should be understood by people with (almost) no knowledge in this field.

In the second chapter we introduce standard bases (generalization of Gröbner bases to non–well–orderings), which are needed for applications to local algebraic geometry (singularity theory), and a method for computing syzygies and free resolutions.

The last chapter describes a new algorithm for computing the normalization of a reduced affine ring and gives an elementary introduction to singularity theory. Then we describe algorithms, using standard bases, to compute infinitesimal deformations and obstructions, which are basic for the deformation theory of isolated singularities.