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, 12:30-1:45, AP&M B-412.

Office hours:

Wednesday 2:00-3:00 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 including affine space over C, R, F_p, Z, Spec's of DVRs, double and multiple points, local schemes, arithmetic surfaces.
• Lecture 3: Sheaf over the base of a topology. The structure sheaf of affine schemes via the distinguished open sets.
• Lecture 4: The stalks of the structure sheaf. Locally ringed spaces. Morphisms of locally ringed spaces. Morphisms of affine schemes. Arbitrary schemes. Closed subschemes. Open subschemes. Schemes associated to prevarities.
• Lecture 5: The proj construction and projective schemes.
• Lecture 6: The functor of points. Fiber product of schemes. Applications: base change, scheme theoretic fiber, scheme theoretic intersections. Examples.
• Lecture 7: Properties of schemes: connected, irreducible, quasicompact, reduced, integral. 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 8: Properties of morphisms. (Affine) local propeties on the target. Isomorphisms, closed immersions, affine morphisms, quasicompact morphisms, separated morphisms, proper morphisms.
• Lecture 9: Finiteness conditions on morphisms, proper morphisms. Sheaves of O_X-modules. Sheaves associated to modules over a ring and over a graded ring. Twisting sheaves. Locally free sheaves.
• Lecture 10: Locally free sheaves and vector bundles. Operations with sheaves. Kernels, cokernels, image, exact sequences. Sheafification.
• Lecture 11: More on sheafification. More on sheaf constructions: direct sums, sheaf hom, duals, tensorization. Operations with vector bundles. Determinants of locally free sheaves.
• Lecture 12: More sheaf constructions: pushforward, pullback, adjoints. Quasicoherent and coherent sheaves.
• Lecture 13: Properties of quasicoherent/coherent sheaves. Kernels, cokernels, images, pullbacks and pushforwards. Extension of sections of quasicoherent sheaves.
• Lecture 14: Serre's theorem on global generation. Sheaf of relative differentials. Affine case and Kahler differentials. Gluing and the general case.
• Lecture 15: Local freeness of the cotangent sheaf and smoothness. Tangent sheaf, canonical line bundle. Geometric genus, plurigenera. The canonical line bundle of the projective space. The Euler sequence. Projective bundles and the generalized Euler sequence.
• Lecture 16: Normal bundles of hypersurfaces. Regular sequences, regular embeddings, local freeness of the normal sheaf. Normal bundle of exceptional divisors in blowups. Divisors. Principal divisors. Rational equivalence. Degree.
• Lecture 17: More on divisors. Line bundles and Cartier divisors. Weil and Cartier divisors over factorial schemes. All line bundles over the projective space.
• Lecture 18: Cohomology of sheaves. Motivation, definition and some examples. Long exact sequence in cohomology.
• Lecture 19: Cech cohomology is independent of the affine cover. Cohomology of line bundles over projective space - statement. Serre's vanishing and finite dimensionality of cohomology.
• Lecture 20: Cohomology of line bundles over projective space. Holomorphic Euler characteristics. Hilbert polynomials. Preview of Math 203c.

Homework:

Homework 1 due January 18 - PDF

Homework 2 due January 30 - PDF

Homework 3 due February 13 - PDF

Homework 4 due February 25 - PDF

Homework 5 due March 6 - PDF

Homework 6 due March 15 - PDF