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.you-know-where.edu, AP&M 6-101.

Lectures: TuTh, 11:00-12:20, AP&M 5-402.

Office hours:

Tu 2-3:30pm 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.

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:

• Lecture notes in algebraic geometry: The Bilkent List

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.

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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: 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 theoretic-intersection. Properties of schemes: reduced, integral, (locally of) finite type, (locally) Noetherian. Affine communication.
• Lecture 7: Affine communication (continued). Factorial varieties. Normal varieties. Normalization.
• Lecture 8: Codim 1 subvarieties in factorial varieties. Singular locus of normal varieties. Normality and Hartogs theorem. Sheaves of O_X-modules. Sheaves attached to modules. Twisting sheaves. Locally free sheaves. Vector bundles.
• Lecture 9: Properties of morphisms (pre-recorded lecture). Isomorphisms, open immersions, closed immersions, quasicompact, quasiseparated, separated, affine, finite type, finite, integral, quasi-finite, proper morphisms.
• Lecture 10: Operations with sheaves: Kernels, cokernels, images for sheaf morphisms. Sheafification. Exact sequences. Direct sum, hom, dual, tensor product. Symmetric and wedge products of locally free sheaves.
• Lecture 11: Pushforward and publlback. Quasi-coherent and coherent sheaves. Definition, examples, first properties.

• Lecture 12: Properties of quasicoherent/coherent sheaves. Kernels, cokernels, images, pullbacks and pushforwards. Extension of sections of quasicoherent sheaves. Serre's theorem.
• Lecture 13: Locally free sheaves and projective modules. Determinants of locally free sheaves. The sheaf of relative differentials. Affine case and Kahler differentials.
• Lecture 14: The sheaf of relative differentials for arbitrary morphisms. Properties.
• Lecture 15: Smootheness and local freeness of the sheaf of differentials. The Euler sequence and generalizations. Projective bundles. Regular sequences. Koszul complex. Regular immersions. Local freeness of the conormal sheaf of regular immersions.
• Lecture 16: Conormal sheaf and various exact sequences. Conormal sheaf for complete intersections, exceptional divisors in blowups. Weil divisors. Principal divisors. Rational equivalence. The sheaf associated to a Weil divisor. Cartier divisors.
• Lecture 17: More on divisors. Line bundles and Cartier divisors. Weil and Cartier divisors over factorial schemes. Divisors of rational sections. All line bundles over the projective space. The ideal sequence of Cartier divisors.
• Lecture 18: Cech cohomology. Some examples. Long exact sequence in cohomology. Cech cohomology is independent of affine cover. Cohomology of line bundles over projective space (statement).
• Lecture 19: Cohomology of line bundles over projective space (proof). Serre's vanishing. Finite dimensionality of cohomology. Holomorphic Euler characteristics.

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, February 14 - PDF

Homework 5 due Thursday, February 23 - PDF

Homework 6 due Tuesday, March 14 - PDF