Welcome to Math 553! This course serves as an introduction to Schemes and Cohomology. Algebraic Geometry is a central subject in modern mathematics, with close connections with number theory, combinatorics, representation theory, differential and symplectic geometry. We will build on Math 552 to study the structure of projective varieties using sheaf cohomology. We will apply the theory to curves and surfaces. We will follow Hartshorne's Algebraic Geometry Chapters 3, 4, 5.

** Lecturer: ** Izzet Coskun, coskun@math.uic.edu

** Office hours: ** MWF 12:50--1:30

** Venue: ** This course will be conducted synchronously on zoom.

** Time: ** MWF 12:00-12:50 pm.

** Text: ** Hartsrhone, Algebraic Geometry

** Grading: ** There will be a takehome midterm and a takehome final problem sets. The grade will be based on these two problem sets. You should solve all the problems in Hartshorne. I will suggest some of the problems that you should definitely try.

** Sylabus: **

Weeks 1-3: Introduction to sheaf cohomology and the cohomology of line bundles on projective space. Review H II.1, II.2, II.5 and read H III.1, III.2, III.3, III. 4, III.5

Weeks 4-5: Ext groups, Serre duality, flatness and smoothness. Review H II.6, II.7, II.8 Read H III.6, III.7, III.8, III.9, III.10

Weeks 6: The semicontinuity theorem. Review H II.9 Read H III.11, III.12

First take home problem set due March 3

Weeks 7-9: Curves, the Riemann-Roch theorem, the canonical embedding. Read H IV 1-6

Weeks 10-11: Surfaces Read H V 1-6.

Weeks 12-15: The construction of the Hilbert scheme

The second take home problem set due April 28.

** Suggested Problems **

Week I: H II.1 Exercises 8, 9, 10, 11, 14, 15, 16, 19, 21

H III.2 Exercises 1, 2, 3, 4, 7

Week II: H II.5 Exercises 1-18

III.4 Exercises 1, 2, 3, 4, 5, 6, 7

III.5 Exercises 1,2,3,4,5,6

Week III: H III.6 Exercises 1, 3, 4, 5, 9, 10

H III.7 Exercises 1, 3, 4

H II.8 Exercises 2, 3, 4, 5, 8

Week IV: H III.8 Exercises 1-4.

H III.9 Exercises 3, 4, 7, 10

Midterm Problem Set Due Wednesday March 3, 2021

Week VI: H IV.1 Exercises 1-8

H IV.2 Exercises 1-5

Week VIII: H IV.3 Exercises 1-6, 12

Week IX H IV.5 Exercises 1-6

H IV.6 Exercises 1-8.

Final Problem Set due Wednesday April 28, 2021

Izzet Coskun Last modified: Tue Feb 2 09:40:02 CST 2021