Math 524 -- Scheme Theory -- Spring 2022

Mondays and Wednesdays 11:30 AM - 12:45 PM on Zoom through January 26 and in MSC N306 after that

This is a graduate-level scheme theory class. We mostly will be following chapters 2 and 3 of Hartshorne. Topics may include sheaves, schemes, separated and proper morphisms, divisors, projective morphisms, sheaf cohomology, and Serre duality. For course details, see the syllabus.

Instructor: Brooke Ullery (bullery@emory, office hours Tuesdays 9-11 AM on Zoom -- same link as the one for class.)

Text:The main text we will use is "Algebraic geometry" by Robin Hartshorne. We will also occasionally use Shafarevich's "Basic algebraic geometry 2."

Homework: I will assign 4-5 problem sets throughout the semester (due every three weeks or so). I'll give you several problems to work on, and some will be graded. You should submit your solutions on canvas. I encourage you to work on the problems together, but you should turn in your own solutions and list the names of your collaborators.

Final project: At the end of the semester, you will choose a recent algebraic geometry paper or preprint and write a 2-3 page summary (more details here). I encourage you to subscribe to daily mailings from arxiv.org in algebraic geometry at the beginning of the semester.

Grading: The final course grade will be calculated as follows: 90% homework/participation, 10% final project. The solutions to the homework problems are mostly readily available online, so it doesn't make much sense for me to assign grades based on them. If you regularly show up to class and turn in homework, you can expect at least an A- on the homework/participation portion of the grade. If you work hard and prove it to me (as evidenced by discussions in class or office hours), you may receive an A.


Assignments

You should submit each problem set on canvas by 11:59 PM the day it's due.

Problem set 1: pdf file, tex file, due February 2
Problem set 2: pdf file, tex file, due February 23
Problem set 3: pdf file, tex file, due March 16
Problem set 4: pdf file, tex file, due April 6
Problem set 5: pdf file, tex file, due April 27

Final project, due May 4

Lecture notes

I'll try to keep these up to date and post them right after we cover each topic in class. However, if we take multiple days to cover a topic, I may not post the notes until we've finished that section.

Section 1: Why schemes? (Jan 10)
Section 2: The Spec of a ring (Jan 10, 12)
Section 3: The Zariski topology (Jan 12, 19)
Section 4: Sheaves (Jan 19, 24)
Section 5: Subsheaves and morphisms of sheaves (Jan 24, 26)
Section 6: The structure sheaf on SpecR (Jan 26, 31)
Section 7: Ringed spaces (Jan 31, Feb 2)
Section 8: Schemes (Feb 2)
Section 9: Graded rings and the Proj construction (Feb 7)
Section 10: Properties of schemes (Feb 7, 9, 14)
Section 11: Fiber products (Feb 14, 16)
Section 12: Separated morphisms (Feb 21, 23)
Section 13: Proper morphisms (Feb 23, 28)
Section 14: Sheaves of modules (Feb 28, Mar 2)
Section 15: Quasi-coherent sheaves (Mar 2, 14)
Section 16: Quasi-coherent sheaves on projective schemes (Mar 16, 21)
Section 17: Global generation of sheaves (Mar 23)
Section 18: Weil divisors (Mar 23, 28, 30, Apr 4)
Section 19: Divisors on curves (Apr 4, 6)
Section 20: Cartier divisors (Apr 6, 11)
Section 21: Invertible sheaves (Apr 11, 13)
Section 22: Morphisms to projective space (Apr 18, 20)
Section 23: Linear systems (Apr 20, 25)
Summer course continuation syllabus