MATH 552 Fall 2018

Algebraic Geometry I

MWF 12 - 1 pm in Addams 303

Course Description

Algebraic geometry is one of the oldest and yet most active disciplines in mathematics, with close connections to number theory, combinatorics, representation theory, complex analysis, differential and symplectic geometry, and it is also used in a wide variety of applied settings. This course is meant to serve as an introduction for graduate students with a view towards future research in a related field. Topics include: affine and projective varieties and schemes, regular and rational maps, function fields, dimension and smoothness, projective curves, and sheaves. We will focus on a mix of both classical examples and modern theory, with the objective that students are subsequently well-prepared to begin a rigorous study of cohomology, intersection theory, moduli, and other advanced topics in the future.

Prerequisites

A familiarity with the topics covered in MATH 516 & 517 (Graduate Algebra I & II) is essential. A background in differential geometry (MATH 553) an commutative algebra (MATH 520) will also be very helpful.

Requirements

There will be three graded components of the course. Homework will be assigned in most lectures (at most three problems a lecture); all of the problems assigned in a given week are due are due at the beginning of class the following Wednesday. Each lecture, one student will be assigned as the scribe. The scribe for a given lecture must write a short (at most one page) recap summarizing the content of the lecture. This should include the statements of all definitions, theorems, lemmas, propositions, and an overview of all examples; it need not, however, include any proofs from the lecture. The summary should be emailed to kftucker@uic.edu as a pdf before the next lecture, and the collection of daily recaps will be posted online. Each student can expect to serve as the scribe at least three times over the course of the semester. Finally, each student is expected to schedule an oral final exam at the end of the semester.

Textbook

Additional references

Homework

Week 1 - Due September 5

Week 2 - Due September 12

Week 3 - Due September 19

Week 4 - Due September 26

Week 5 - Due October 3

Week 6 - Due October 10

Week 7 - Due October 17

Week 8 - Due October 24

Week 9 - Due October 31

Week 10 - Due November 7

Week 11 - Due November 14

Week 12 - Due November 21

Week 13 - Due November 28

Week 14 - Due December 5

Week 15 - Due December 12