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: M 9-11, W 9-10 and by appointment in SEO 423
Venue: Taft 321
Time: MWF 12:00-12:50 pm.
Text Hartsrhone, Algebraic Geometry
Grading I will regularly assign homework. There will be a takehome midterm and takehome final problem sets. The grade will be based on these two problem sets.
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 4-6: 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 7: The semicontinuity theorem. Review H II.9 Read H III.11, III.12
First take home problem set due March 7
Weeks 8-10: Curves, the Riemann-Roch theorem, the canonical embedding. Read H IV 1-6
Weeks 11-15: Surfaces Read H V 1-6.
The second take home problem set due May 1.
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 February 28, 2020 pdf
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 April 29, 2020 pdf