Math 562: Intro to Algebraic Geometry 2

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Catalog Information

Title

Introduction to Algebraic Geometry 2.

Credit Hours

3

Prerequisite

Math 561 or concurrent enrollment.

Description

Local properties of quasi-projective 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 non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor.

  1. Local properties of algebraic varieties
    • The local ring at a point
    • Zariski tangent space
    • Singular points
    • The tangent space
  2. Power series expansions
    • Local parameters
    • The completion of a local ring
  3. Properties of nonsingular points
  4. Birational maps
    • Blowup in projective space
    • Local blowup
    • Behavior of a subvariety under a blowup
  5. Normal varieties and normalization
  6. Divisors
    • Cartier divisors
    • Weil divisors
  7. 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