Math 216A: Algebraic Geometry

 Fall 2024

TuTh 12:00-01:20 PM, Room 381T.
Solving equations with given coefficients is a fundamental topic in mathematics. We may use linear algebra to solve linear equations and find applications in the real world. In general, polynomial equations are hard to solve (and there may be infinitely many solutions). Instead, in algebraic geometry we study \textbf{invariants and geometry} of the space of solutions to polynomial equations. These invariants (e.g. dimensions, structure sheaf of functions, tangent spaces) have geometric meanings and could also be understood using commutative algebra. Geometric intuition (from complex and real numbers) could be used to predict behaviors of algebraic invariants beyond intuition (e.g. over finite fields) and vice versa. In its modern reformulation based on the concept of a scheme, the subject has acquired awesome technical power and its techniques not only permit a better arsenal with which to study classical problems over any field (not just $\BC$), but also have a vast range of applicability beyond the classical concerns: algebraic methods for studying analytic concepts, a geometric foundation that allows one to "visualize" commutative algebra and number theory, a source of important constructions and techniques in representation theory, a common framework in which one can view Galois theory and fundamental groups as "the same thing", and so on ad infinitum. The scheme theory has some important advantages: the notion of fibered products (tensor products of rings and flatness), deformation theory (nilpotent thickenings) and coherent sheaves (linear algebra over any ring), cohomology theory (Serre duality and derived functors), moduli spaces (geometry of the collection of all objects), pullback and pushforward of sheaves (relations of different spaces via maps between them).

Instructor: Zhiyu Zhang (zyuzhang@stanford.edu).

Office Hour: Monday 9:30-11 AM, and by appointment.

Prerequisites:

Textbook: Algebraic Geometry by Hartshorne (required), Commutative Ring Theory by Matsumura (recommended). See also the webpage of Conrad's course.

Grade: 100% Homework.

Homework (Due Friday 11 am each week):

  • Problem Set 1
  • Problem Set 2
  • Problem Set 3
  • Problem Set 4
  • Problem Set 5
  • Problem Set 6
  • Problem Set 7
  • Problem Set 8
  • Problem Set 9

    Additional exercise (optional): Problem sets from Bhatt's course; Exercise chapter in Stacks project.

    Lecture Schedule

  • Lecture 1, Sep 24.
  • Lecture 2, Sep 26.
  • Lecture 3, Oct 1. Zoom class.
  • Lecture 4, Oct 3. Zoom class.
  • Lecture 5, Oct 8.
  • Lecture 6, Oct 10.
  • Lecture 7, Oct 15.
  • Lecture 8, Oct 17.
  • Lecture 9, Oct 22.
  • Lecture 10, Oct 24.
  • Lecture 11, Oct 29.
  • Lecture 12, Oct 31.
  • Lecture 13, Nov 5. No class (Democracy Day).
  • Lecture 14, Nov 7.
  • Lecture 15, Nov 12.
  • Lecture 16, Nov 14.
  • Lecture 17, Nov 19.
  • Lecture 18, Nov 21.
  • Lecture 19, Nov 26. No class (Thanksgiving Recess).
  • Lecture 20, Nov 28. No class (Thanksgiving Recess).
  • Lecture 21, Dec 3.
  • Lecture 22, Dec 5.

    Access and Accommodations
    Stanford is committed to providing equal educational opportunities for disabled students. Disabled students are a valued and essential part of the Stanford community. We welcome you to our class.
    If you experience disability, please register with the Office of Accessible Education (OAE). Professional staff will evaluate your needs, support appropriate and reasonable accommodations, and prepare an Academic Accommodation Letter for faculty. To get started, or to re-initiate services, please visit oae.stanford.edu.