Algebraic Geometry I

This volume of the Encyclopaedia consists of two parts. The first is devoted to the theory of curves, which are treated from both the analytic and algebraic points of view. Starting with the basic notions of the theory of Riemann surfaces the reader is lead into an exposition covering the Riemann-Roch theorem, Riemann's fundamental existence theorem, uniformization and automorphic functions. The algebraic material also treats algebraic curves over an arbitrary field and the connection between algebraic curves and Abelian varieties. The second part is an introduction to higher-dimensional algebraic geometry. The author deals with algebraic varieties, the corresponding morphisms, the theory of coherent sheaves and, finally, the theory of schemes. This book is a very readable introduction to algebraic geometry and will be immensely useful to mathematicians working in algebraic geometry and complex analysis and especially to graduate students in these fields. Keywords: Algebraic geometry, algebraic curves, Riemann surfaces, sheaves, schemes, Riemann-Roch formula, algebraic variety Contents: Contents: Riemann Surfaces and Algebraic Curves by V.V. Shokurov.- Algebraic Varieties and Schemes by V.I. Danilov.