Math 203A - Algebraic Geometry

Welcome to Math 203b!

Course description:

This course provides an introduction to algebraic geometry. Algebraic geometry is a central subject in modern mathematics, and an active area of research. It has connections with number theory, differential geometry, symplectic geometry, mathematical physics, string theory, representation theory, combinatorics and others.

Math 203 is a three quarter sequence. Math 203b will cover schemes, sheaves and cohomology.

Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu, AP&M 6-101.

Lectures: TuTh, 11:00-12:20, AP&M 5-402.

Office hours:

Tu 2-3:30pm in AP&M 6-101. The office hours will be hybrid, the zoom link can be found in Canvas

I am available for questions after lecture or by appointment. Also, feel free to drop in if you see me in my office.

Textbook: I will roughly follow Andreas Gathamnn's notes available online. I recommend that you also consult Hartshorne's Algebraic Geometry or Vakil's The Rising Sea.

There will be no exams for this class. The grade will be based entirely on homeworks and regular attendance of lectures. The problem sets are mandatory and are a very important part of the course. The problem sets are submitted via Gradescope.

Important dates:

• First class: Tuesday, January 10.
• University Holiday: Monday, January 16.
• University Holiday: Monday, February 20.
• Last class: Thursday, March 16.

Annoucements:

The lectures on February 8, February 10 will be pre-recorded.

Lecture Summaries

• Lecture 1: Motivation for schemes. The prime spectrum. Zariski topology - Notes
• Lecture 2: Some examples of prime spectra including affine space over C, R, F_p, Z, Spec's of DVRs, double and multiple points, local schemes, arithmetic surfaces.
• Lecture 3: Sheaves over the base of a topology. The structure sheaf of affine schemes via the distinguished open sets.The stalks of the structure sheaf. Locally ringed spaces. Morphisms of locally ringed spaces.
• Lecture 4: Morphisms of affine schemes. Open and closed subschemes. Schemes. Prevarieties and schemes. Proj construction.
• Lecture 5: More on Proj. The functor of points. Fiber products.
• Lecture 6: Constructions involving fiber product: base change, fibers, scheme theoretic-intersection. Properties of schemes: reduced, integral, (locally of) finite type, (locally) Noetherian. Affine communication.
• Lecture 7: Affine communication (continued). Factorial varieties. Normal varieties. Normalization.

Homework:

Homework 1 due Tuesday, January 17 - PDF

Homework 2 due Tuesday, January 24 - PDF

Homework 3 due Tuesday, January 31 - PDF

Homework 4 due Tuesday, January 14 - PDF