Difference between revisions of "Math 562: Intro to Algebraic Geometry 2"
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Revision as of 15:42, 3 April 2013
Contents
Catalog Information
Title
Introduction to Algebraic Geometry 2.
Credit Hours
3
Prerequisite
Math 671 or concurrent enrollment.
Description
Local properties of quasiprojective varieties. Divisors and differential forms.
Desired Learning Outcomes
Prerequisites
Math 561
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.
 Local properties of algebraic varieties
 The local ring at a point
 Zariski tangent space
 Singular points
 The tangent space
 Power series expansions
 Local parameters
 The completion of a local ring
 Properties of nonsingular points
 Birational maps
 Blowup in projective space
 Local blowup
 Behavior of a subvariety under a blowup
 Normal varieties and normalization
 Divisors
 Cartier divisors
 Weil divisors
 Differential forms
Textbooks
Possible textbooks for this course include (but are not limited to):
I. R. Shararevich, Basic Algebraic Geometry I, Varieties in Projective Space
Additional topics
As this is a terminal course, it may be possible to substitute other topics for the above, especially items 6 and 7. Some instructors may wish to give an overview on the moduli of curves and its relation to mathematical physics.
Courses for which this course is prerequisite
None