Algebraic Varieties

One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on curves, gleaned in part through the theorems of Mordell, Weil, Siegel, and Faltings. It has become clear that the study of rational and integral points has deep connections to other branches of mathematics: complex algebraic geometry, Galois and étale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory.

The articles in this volume study various cohomological aspects of algebraic varieties:
- characteristic classes of singular varieties;
- geometry of flag varieties;
- cohomological computations for homogeneous spaces;
- K-theory of algebraic varieties;
- quantum cohomology and Gromov-Witten theory.

This EMS volume consists of two parts. The first part is devoted to the exposition of the cohomology theory of algebraic varieties. The second part deals with algebraic surfaces. The authors, who are well-known experts in the field, have taken pains to present the material rigorously and coherently. The book contains numerous examples and insights on various topics. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

This volume contains the proceedings of the Algebraic Geometry Conference on Classification of Algebraic Varieties, held in May 1992 at the University of L'Aquila in Italy. The papers discuss a wide variety of problems that illustrate interactions between algebraic geometry and other branches of mathematics.

I. Algebraic Varieties in a Projective Space.- I. Fundamental Concepts.- § 1. Plane Algebraic Curves.- 1. Rational Curves.- 2. Connections with the Theory of Fields.- 3. Birational Isomorphism of Curves.- Exercises.- §2. Closed Subsets of Affine Spaces.- 1. Definition of Closed Subset.- 2. Regular Functions on a Closed Set.- 3. Regular Mappings.- Exercises.- § 3. Rational Functions.- 1. Irreducible Sets.- 2. Rational Functions.- 3. Rational Mappings.- Exercises.- § 4. Quasiprojective Varieties.- 1. Closed Subsets of a Projective Space.- 2. Regular Functions.- 3. Rational Functions.