M 672 Fall 2002

Projective Geometry I

(An introduction to Algebraic Geometry)

Instructor: Rick Miranda

MWF 4-5pm, EE203

Reference number: 286891

Text: Basic Algebraic Geometry I: Varieties in Projective Space,

by I.R. Shafarevich

Course Outline:

I. Algebraic Sets

Affine space, projective space, coordinates, affine algebraic sets, projective algebraic sets, basic examples

II. Algebraic Aspects of Geometry

Ideals of algebraic sets, irreducibility, the Hilbert Basis Theorem, the Nullstellensatz, applications, the Zariski topology

III. Polynomial Functions

Coordinate rings, rational functions, regular functions, presheaves and sheaves of functions

IV. Regular and Rational Maps

Basic definitions, isomorphisms, birational isomorphisms, projections, examples

V. Fundamental properties of varieties

Degree, dimension, first theorems, examples

VI. Local properties of varieties

Smoothness, singularities, tangent spaces

VII. More Sheaf Theory

Maps of sheaves, introduction to cohomology of sheaves

There will be constant homework, and no examinations. Grades will be based on class participation and homework completeness.