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. For Math 203B, the tentative plan is to cover schemes and cohomology. This corresponds roughly to Chapters 2 and 3 of Hartshorne.

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

Lectures: WF, 10:00-11:15, AP&M B-412.

Office hours:

Wednesday 11:30-12:30 in AP&M 6-101.

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

Additional resources:

• Lecture notes in algebraic geometry: The Bilkent List

Prerequisites:

Math 203A. Some knowledge of modern algebra at the level of Math 200 is required. I will try to keep the algebraic prerequisites to a minimum. Familiarity with basic point set topology, complex analysis and/or differentiable manifolds is helpful to get some intuition for the concepts. Since it is hard to determine the precise background needed for this course, I will be happy to discuss prerequisites on an individual basis. If you are unsure, please don't hesitate to contact me.

Grading:

There will be no exams for this class. The grade will be based entirely on homeworks and regular attendance of lectures. It is impossible to learn the subject without solving the problem sets. The problem sets are due in class.

Important dates:

Lecture Summaries

• Lecture 1: Motivation for schemes. The prime spectrum. Zariski topology.
• Lecture 2: Some examples of prime spectra. Strategy for constructing the structure sheaf.
• Lecture 3: Sheaf on the base of a topology. The structure sheaf over the prime spectrum. The stalks of the structure sheaf.
• Lecture 4: Locally ringed spaces. Morphisms of affine schemes. Open and closed subschemes. Schemes. Prevarieties and schemes. Proj construction.
• Lecture 5: The functor of points. Fiber products. Base change. Fibers. Scheme theoretic intersections. Properties of schemes: connected, irreducible, quasicompact, reduced, integral.
• Lecture 6: Integral means reduced and irreducible. Further properties of schemes: (locally of) finite type, (locally) Noetherian schemes. Arbitrary open sets inheriting properties from the open cover.
• Lecture 7: Properties of morphisms: isomorphism, closed immersions, affine, quasicompact, locally of finite type, finite, quasi-finite, separated, proper. Checking properties (affine)-locally on the target.
• Lecture 8: Sheaves of O_X-modules. Examples. Twisting sheaves over projective schemes. Locally free sheaves and vector bundles. Kernels and cokernels for sheaf morphisms. Sheafification.
• Lecture 9: More on sheafification. Constructions with sheaves: direct sum, hom, dual, tensor product. Symmetric and wedge products of locally free sheaves. Direct and inverse images. Examples.
• Lecture 10: Quasi-coherent and coherent sheaves. Kernels, images, cokernels, direct and inverse images of quasi-coherent/coherent sheaves. Extension properties of sections of coherent sheaves.
• Lecture 11: Global generation and Serre's theorem. Sheaf of (relative) differentials. Affine case. Global construction.
• Lecture 12: Cotangent sheaf and smoothness. Examples: Euler sequence over projective space and generalizatons.
• Lecture 13: Canonical bundles of hypersurfaces. Line bundles and divisors. Weil divisors. Principal divisors. Rational equivalence. Some examples.
• Lecture 14: More on divisors. Cartier divisors and line bundles. Weil and Cartier divisors over factorial schemes. All line bundles over projective space.
• Lecture 15: Taking global sections is not right exact. Cech cohomology. Some examples. Long exact sequence in cohomology.
• Lecture 16: Cech cohomology is independent of affine cover. Cohomology of line bundles over projective space (statement). Finiteness of cohomology. Serre vanishing. Euler characteristic.
• Lecture 17: Cohomology of projective space. Riemann-Roch for curves. Statement of Serre duality and equality of the genera.
• Lecture 18: Examples of genus calculuations: plane curves, nodal curves. Topological genus. Riemann-Hurwitz: algebraic and topological version.
• Lecture 19: Kodaira vanishing theorem for curves. Table of cohomology groups for curves. Projective embeddings: basepoint free, ample, very ample. Examples for projective spaces and for curves.
• Lecture 20: Criterion for proving global generation and ampleness for curves. The canonical morphism. Hyperelliptic curves. Classification of curves of genus 0, 1, 2, 3, 4, 5.

Homework:

Homework 1 due Friday, January 19 - PDF

Homework 2 due Monday, January 29 - PDF

Homework 3 due Wednesday, February 7 - PDF

Homework 4 due Friday, February 16 - PDF

Homework 5 due Friday, February 23 - PDF

Homework 6 due Friday, March 9 - PDF

Homework 7 due Friday, March 16 - PDF