Math 819

Math 819, Topics in Algebraic Geometry: Schemes

Texts:
The Rising Sea: Foundations of Algebraic Geometry, by Ravi Vakil (available online here)
Algebraic Geometry, by Hartshorne
The Geometry of Schemes, by David Eisenbud and Joe Harris (reference)

Time: TuTh 10:30-12:20
Location: AQ 5050
Office hours: M 3:30-4:30pm and Tu 4:30-5:30pm (or by appointment)

Syllabus: We will plan to cover approximately Chapter 2 and (time permitting) part of Chapter 3 of Hartshorne. For certain parts of the material, we will follow Vakil. Homework will consist of problems from either of these two books.

Final exam: April 19, noon-3pm in AQ 5005.

The Axioms: I firmly believe in the axioms laid out by Federico Ardila.

Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Everyone can have joyful, meaningful, and empowering mathematical experiences.

Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Every student deserves to be treated with dignity and respect.

HW:
HW1 (sheaves)
HW2 (schemes) (partial solutions)
HW3 (local properties) (partial solutions)
HW4 (fiber products and Proj) (partial solutions)
HW5 (separated and proper; locally free sheaves) (partial solutions)
HW6 (divisors, line bundles, maps to projective space)

Daily schedule (topics from class, lecture notes, homework):

Week Topics

1: [1/05] Basics (why schemes?). Gluing topological spaces.

2: [1/10] Presheaves and sheaves.
[1/12] No class (Math Dept all-day meeting)

3: [1/17] Sheafification and universal properties.
[1/19] Sheaves on a base; start the structure sheaf on Spec R.

4: [1/24] Finished the structure sheaf of Spec R. Definition of scheme.
[1/26] Examples of schemes: gluing, generic points, A^1 over various fields.

5: [1/31] Examples of schemes cont'd: regular functions, nilpotents, reducedness. Connected, irreducible.
[2/02] Modules, O_X-modules, Noetherianness.

6: [2/07] Quasicoherence. Affine-local properties. The Affine Communication Lemma. Finiteness properties.
[2/09] Quasicoherence is local. Started morphisms.

7: [2/14] Morphisms of schemes and "families". Kinds of morphisms: affine, projective, open/closed embeddings
[2/16] More kinds of morphisms: quasicompact, (locally) finite type. Started tensor products.

8: [2/21] No class (reading break)
[2/23] No class (reading break)

9: [2/28] Fiber products (Products, intersections, fibers / preimages, change of base)
[3/02] Proj (finally)

10: [3/07] Proj continued. Maybe start separated and proper morphisms.
[3/09] Separated morphisms

11: [3/14] Proper morphisms
[3/16] Global Spec, Global Proj. Started vector bundles.

12: [3/21] Vector bundles and locally free sheaves.
[3/23] Locally free sheaves, cont'd.

13: [3/28] Weil divisors, class groups.
[3/30] Normality.

14: [4/04] Cartier divisors and line bundles.
[4/06] Line bundles and maps to projective space. (Afternoon: smoothness and the canonical bundle)

15: [4/11] Line bundles and maps. Functor of points of projective space. Ample, very ample and big divisors.
(Afternoon: Intro to sheaf cohomology.)

Week	Topics
1:	[1/05] Basics (why schemes?). Gluing topological spaces.
2:	[1/10] Presheaves and sheaves. [1/12] No class (Math Dept all-day meeting)
3:	[1/17] Sheafification and universal properties. [1/19] Sheaves on a base; start the structure sheaf on Spec R.
4:	[1/24] Finished the structure sheaf of Spec R. Definition of scheme. [1/26] Examples of schemes: gluing, generic points, A^1 over various fields.
5:	[1/31] Examples of schemes cont'd: regular functions, nilpotents, reducedness. Connected, irreducible. [2/02] Modules, O_X-modules, Noetherianness.
6:	[2/07] Quasicoherence. Affine-local properties. The Affine Communication Lemma. Finiteness properties. [2/09] Quasicoherence is local. Started morphisms.
7:	[2/14] Morphisms of schemes and "families". Kinds of morphisms: affine, projective, open/closed embeddings [2/16] More kinds of morphisms: quasicompact, (locally) finite type. Started tensor products.
8:	[2/21] No class (reading break) [2/23] No class (reading break)
9:	[2/28] Fiber products (Products, intersections, fibers / preimages, change of base) [3/02] Proj (finally)
10:	[3/07] Proj continued. Maybe start separated and proper morphisms. [3/09] Separated morphisms
11:	[3/14] Proper morphisms [3/16] Global Spec, Global Proj. Started vector bundles.
12:	[3/21] Vector bundles and locally free sheaves. [3/23] Locally free sheaves, cont'd.
13:	[3/28] Weil divisors, class groups. [3/30] Normality.
14:	[4/04] Cartier divisors and line bundles. [4/06] Line bundles and maps to projective space. (Afternoon: smoothness and the canonical bundle)
15:	[4/11] Line bundles and maps. Functor of points of projective space. Ample, very ample and big divisors. (Afternoon: Intro to sheaf cohomology.)

819 final projects:

I have decided the final project will be optional -- You may instead choose to do the last HW assignment (which will be HW 6).

Description: Expository writeup on a topic of your choice. Your sources can be expository AG writing (including Hartshorne and Vakil), survey articles, or other expository sources.

Format:

Approximately 5 pages in LaTeX amsart format.
Your writeup must include at least one illustrative example or concrete computation, not just abstract exposition and general theorems. It is OK to omit details and proofs that are beyond the scope of your writeup.
The audience is a fellow student in the course -- not the instructor.

You must pick a topic you don't already know about. You are encouraged, however, pick a topic that is relevant to your research interests.

List of suggested topics:

Grassmannians and Schubert varieties -- Fulton's Young Tableaux or Eisenbud and Harris, 3264 and All That
Hilbert polynomials and the Hilbert scheme
toric varieties
abelian varieties (theorem of the square, theorem of the cube, see-saw principle etc -- proof that a proper algebraic group must be abelian) -- Milne's "Abelian Varieties"
curves (e.g. Hartshorne Ch. 4 Sections 1-3)
ruled surfaces (e.g. Hartshorne Ch. 5 Section 1-2)
Chern classes
GAGA (Serre's original paper is quite readable if you black-box the background results of complex geometry and commutative algebra)
Any other topic of interest, but consult with me first.

Assessment will be based on:

Clarity of exposition and of mathematics
Quality of examples / computations
Appropriateness for the intended audience (fellow Math 819 students)