Math 203B - Algebraic Geometry
Welcome to Math 203B!
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 two quarter sequence. Math 203b will cover vector
bundles, sheaf cohomology, algebraic curves, Riemann-Roch, Serre
duality. Other possible topics: Chow groups, Chern classes,
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: MW, 2pm-3:20pm, AP&M 7-421 (will most likely
- Monday 3:30-4:30, Wednesday 1-2.
Textbook: There is no required textbook. I will follow
Andreas Gathamnn's notes available online
lecture or by appointment. Also, feel free to drop in if you see me in my
Other useful texts are
- Igor Shafarevich, Basic Algebraic Geometry I, Varieties in
- Joe Harris, Algebraic Geometry: a first course.
- David Mumford, Algebraic Geometry I, Complex Projective
More advanced but useful references are:
- Robin Hartshorne, Algebraic
- David Mumford, The red book of varieties and schemes.
- There will be no exams or homeworks for this
- Lecture 1: Sheaves of
O_X-modules. Locally free sheaves and vector bundles. Functorial
properties of sheaves. Kernels. Cokernels. Sheafification. Short exact
- Lecture 2: More on sheafification and short exact sequences.
Direct sums, tensor products, determinants, wedge products and
other natural operations with vector bundles and
- Lecture 3: Quasi-coherent sheaves. Functorial
constructions. Ideal sheaves. Pullbacks.
- Lecture 4: The sheaf of differentials.
- Lecture 5: Examples: the canonical sheaf of P^n and
hypersurfaces in P^n.
- Lecture 6: Divisors and rational equivalence. Line bundles
associated to divisors. Statemenet of Riemann-Roch.
- Lecture 7: Riemann-Hurwitz (algebraic and topological
versions). Ramification divisors. The degree of the canonical divisor.
- Lecture 8: Proof of Riemann-Roch via the residue theorem.
- Lecture 9: Applications of Riemann-Roch. Morphisms to
projective space; globally generated
and very ample line bundles. Examples.
- Lecture 10: More applications of Riemann-Roch: genus 0 curves
are isomorphic to P^1. Genus 1 curves are cubics. Addition law on genus 1
curves. Genus 2 curves and hyperelliptic curves. Canonical embedding of
- Lecture 11: Cohomology of sheaves. Cech cohomology.
- Lecture 12: Independence of affine cover. Long exact sequence
- Lecture 13: Kodaira vanishing. More on Riemann-Roch. Serre
- Lecture 14: Cohomology of projective space. Serre's theorems.
- Lecture 15: Intersection theory. Chow groups. The excision
sequence. Chow group of P^n. Weil and
- Lecture 16: Line bundles and Cartier divisors. Intersections
with Cartier divisors. Intersection
pairing on surfaces. Curves on surfaces and their genus.
- Lecture 17: Riemann-Roch for surfaces.
Hirzebruch-Riemann-Roch stated. Axiomatics of Chern
classes. Todd genus. Chern character.
- Lecture 18: Construction of Chern classes. Applications.