Welcome to Math 203c!

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 203C, the tentative plan is to finish the discussion of curves started at the end of 203b, then cover Chern classes, intersection theory, and Hirzebruch-Riemann-Roch.

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 203AB. 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: Catch-up from Math 203b. Classification of low genus curves. Elliptic curves.
• Lecture 2: Serre duality. Ideas behind the argument. The existence of a dualizing sheaf.
• Lecture 3: Proof of Serre duality continued.
• Lecture 4: Families of curves. General introduction to flatness. Some useful algebraic facts.
• Lecture 5: More on flatness. Some examples and geometric visualization. Invariants preserved by flatness: Euler characteristic, dimension, degree, genus.
• Lecture 6: Introduction to moduli problems. Hilbert scheme. Applications of flatness: infinitesimal deformations and tangent space to the Hilbert scheme. Some examples.
• Lecture 7: Intersection theory. Motivation for Chow groups and some goals. Definition of Chow groups. Rational equivalence of cycles.
• Lecture 8: Some examples of Chow groups. Localization sequence. Affine bundles.
• Lecture 9: Affine stratifications. Proper morphisms. Proper pushforwards. Examples of Chow groups: projective spaces, grassmannians, blowups.
• Lecture 10: Proper morphisms preserve rational equivalence. General discussion of intersection products.
• Lecture 11: Cartier divisors. Intersection with Cartier divisors. Projection formula. Applications: genus of curves on surfaces, Chow rings of blowups.
• Lecture 12: Commutativity of intersection products. Segre and Chern classes defined. First properties of Segre and Chern classes.
• Lecture 13: Whitney's formula. Chern classes of projective spaces, hypersurfaces in projective spaces. Splitting principle. Some examples.
• Lecture 14: Chern roots. Geometric interpretation of Chern classes. Application: 27 lines on a cubic surface.
• Lecture 15: Chow groups of vector bundles and projective bundles. Chern character. Todd genus. Statement of HRR. Examples: curves, surfaces.
• Lecture 16: HRR for line bundles over projective space. Resolutions by vector bundles and HRR for projective space. K-theory of smooth projective varieties.
• Lecture 17: Homological algebra of vector bundle resolutions. Chern character of sheaves via resolutions. Statement for GRR for inclusions, and how it implies HRR for arbitrary smooth varieties.
• Lecture 18: GRR for inclusions. Comparison of K-theory and Chow. A particular case of GRR for inclusions: sections of projective bundles.
• Lecture 19: Deformation to the normal cone. Finishing the proof of GRR for inclusions. GRR for arbitrary morphisms.
• Lecture 20: Applications of GRR. The tautological ring of the moduli of curves: kappa and lambda classes. The canonical class for the moduli of curves.

Homework:

Homework 1 due Wednesday, April 18 - PDF

Homework 2 due Wednesday, May 2 - PDF

Homework 3 due Friday, May 11 - PDF

Homework 4 due Friday, May 18 - PDF

Homework 5 due Friday, June 1 - PDF