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.youknowwhere.edu,
AP&M 6101.
Lectures: TuTh, 11:0012:20, AP&M 5402.
Office hours:
 Tu 23:30pm in AP&M 6101. 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.
Additional resources:
Grading:  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 prerecorded.
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 theoreticintersection. 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