Special Topics in

ALGEBRAIC GEOMETRY 2

Math 689-605

Spring 1994

Instructor: Peter Stiller
Office: Milner Hall 215
Office Phone: 5-5727
E-mail: stiller@alggeo.tamu.edu
Office Hours: By appointment

Time: Tuesdays and Thursdays 12:45 to 2:00
Place: Acad. 206
Text: Course Notes will be provided.
Supplementary Material: See bibliography below
Prerquisites: The equivalent of Math 416 or Math 654 (Rings and Modules) and background in Linear Algebra.

Overview: Algebraic Geometry is a subject with historical roots in analytic geometry. At its most naive level it is concerned with the geometry of the solutions of a system of polynomial equations. In its early days the subject developed around the classification problem, the search for invariants of transformations, intersection problems, and the study of families of points on a curve or curves on a surface (known as linear systems). It made use of techniques from geometry (projective geometry), number theory (Diophantine equations), and analysis (elliptic and abelian integrals). Today it is a powerful synthesis of those algebraic, geometric, and analytic techniques. Results can be universally applied to a range of problems from the discrete (such as the recent proof of Fermat's Last Theorem) to the continuous (global complex analysis). It subsumes most of commutative algebra and much of algebraic number theory, and overlaps with differential geometry, modern "analytic geometry" (complex manifolds), Lie groups, representation theory, theoretical physics, and to a lesser extent the theory of partial differential equations. In addition to being one of the central disciplines of pure mathematics, algebraic geometry has developed an applied side which is linked to problems in computational complexity and the theory of algorithms, symbolic computation, robotics, control theory, computational geometry, geometric modeling, image recognition, computer vision, and scientific visualization.

Syllabus:

1/18 Review of Properties of Affine Algebraic Sets
1/20 Review of Nullstellensatz, Module Finiteness Conditions, Integral Ring Extensions, Field Theory
1/25 Review of Varieties and Maps
1/27 Rational Functions, Local Rings, and Discrete Valuation Rings
2/1 Ideals, Exact Sequences, and Free Modules
2/3 Plane Curves - Local Properties
2/8 Multiplicities
2/10 Review of Projective Varieties
2/15 Problems Set #1 due and discussed
2/17 Introduction to Projective Plane Curves

Basis for Grade: Three Problem Sets (25% each), Take-Home Final (25%)

Bibliography:

Hartshorne, "Algebraic Geometry", Springer GTM #52

Griffiths and Harris, "Principles of Algebraic Geometry" Wiley Interscience

Shafarevich, "Basic Algebraic Geometry", Springer

Kirwan, "Complex Algebraic Curves", Cambridge Univ. Press

Silverman, "The Arithmetic of Elliptic Curves" Springer GTM #106

Fulton, "Algebraic Curves" Benjamin 1969

Assignment #1 (1/18 and 1/20) Read the handout on affine algebraic sets, pages 1-33. This should mostly be review. Look over the exercises and try any that look interesting.

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