This book and the following second volume is an introduction into modern
algebraic geometry. In the first volume the methods of homological
algebra, theory of sheaves, and sheaf cohomology are developed. These
methods are indispensable for modern algebraic geometry, but they are
also fundamental for other branches of mathematics and of great interest
in their own.
In the last chapter of volume I these concepts are applied to the theory
of compact Riemann surfaces. In this chapter the author makes clear how
influential the ideas of Abel, Riemann and Jacobi were and that many of
the modern methods have been anticipated by them.
For this second edition the text was completely revised and corrected.
The author also added a short section on moduli of elliptic curves with
N-level structures. This new paragraph anticipates some of the
techniques of volume II.