Difference between revisions of "Math 561: Intro to Algebraic Geometry 1"
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Revision as of 15:42, 3 April 2013
Contents
Catalog Information
Title
Introduction to Algebraic Geometry 1.
Credit Hours
3
Prerequisite
Math 671 or concurrent enrollment.
Description
Basic definitions and theorems on affine, projective, and quasiprojective varieties.
Desired Learning Outcomes
Prerequisites
Math 671 or concurrent enrollment.
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means they should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and nonexamples of the various concepts. The students should be able to demonstrate their mastery by solving nontrivial problems related to these concepts, and by proving simple (but nontrivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.
 Algebraic plane curves
 Rational curves
 Relation with field theory
 Rational maps
 Singular and nonsingular points
 Projective spaces
 Affine varieties
 Affine space and the Zariski topology
 Regular functions
 Regular maps
 Rational functions and rational maps
 Quasiprojective varieties
 The Zariski topology on projective space
 Regular and rational functions
 Examples
 Products and maps of quasiprojective space
 Definition of products
 Properness of projective maps
 Finite maps
 Normalization
 Dimension
 Definition of dimension
 Dimension of intersection with a hypersurface
 Dimension of fibres
 Application to lines on surfaces (optional)
Textbooks
Possible textbooks for this course include (but are not limited to):
 I. R. Shafarevich, Basic Algebraic Geometry 1, Varieties in Projective Space
Additional topics
If time permits, additional topics may be covered. Possibilities include the 27 lines on a cubic surface, or an introduction to elliptic curves.