代数几何ALGEBRAIC GEOMETRY

In 1989-1990 I taught a course in Algebraic Geometry at Stanford Univer-sity, writing up lecture notes. These were revised for publication in 1998. In 1989-90 I covered the material in Chapters 1-14 in two quarters, and con-tinued with a quarter on cohomology of coherent sheaves, lecturing out of Hartshorne's book. The aim is to make this a text that can be used in a one year at the graduate level. I have tried to give complete proofs assuming a background in algebra at the level one expects from a first or second year graduate student. The point of view here is that of Serre [23] or Chapter I of Mumford [21]--a variety is a ringed space locally isomorphic to an affine variety over a field, which is algebraically closed except in Chapter 14. Although I do not treat schemes I trust the reader will not find the transition too difficult. The first eight sections contain material applicable to varieties of every dimension, the last six contain material which is particular to the theory of curves. We give a portion of the general theory of elliptic curves, the zeta function of a curve and Riemann hypothesis. For most of the book I consider irreducible varieties over an algebraically closed field. In Chapter 14, we work over a finite field.