Math 203A - Algebraic Geometry

Welcome to Math 203a!

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 two quarter sequence. Math 203a (Fall Quarter) will be cover affine and projective varieties (roughtly the first 2/3 of the quarter) and basics of scheme theory (the last 1/3 of the quarter).

The Winter Quarter will cover more scheme theory, sheaf cohomology, algebraic curves, Riemann-Roch, Serre duality.

The course description can be found here .

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

Lectures: MW, 2pm-3:20pm, AP&M 7-421 (will most likely change).

Office hours:

Tuesday 2-4 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: There is no required textbook. I will follow Andreas Gathamnn's notes available online .

Other useful texts are

• Igor Shafarevich, Basic Algebraic Geometry I, Varieties in Projective Space
• Joe Harris, Algebraic Geometry: a first course.
• David Mumford, Algebraic Geometry I, Complex Projective Varieties

More advanced but useful references are:

• Robin Hartshorne, Algebraic geometry
• David Mumford, The red book of varieties and schemes.
Notes: There may be typos in the files below, let me know if you spot any serious ones.
• Complete notes from a course taught in Spring 2008, courtesy of David Philipson PDF.
• Notes on resultants PDF.
• Notes on irreducibility and dimension PDF.

Prerequisites:

Some knowledge of modern algebra at the level of Math 200 is required. However, I will not assume background in commutative algebra. Familiarity with complex analysis, basic point set topology, differentiable manifolds is helpful, but not required. 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.

There will be no exams for this class. The grade will be based entirely on homeworks. The problem sets will be posted online.

Important dates:

• Veterans Day: November 11.
• Thanksgiving break: November 26-27.
• Last day of classes: December 4.

Lecture Summaries

• Lecture 1: Affine algebraic sets. Noetherian rings. Zariski topology.
• Lecture 2: Ideals of affine algebraic sets. Hilbert's Nullstellensatz. Weak form implies strong NS. Started the proof of the weak NS. Resultants.
• Lecture 3: Finished the proof of the weak Nullstellensatz. Irreducibility.
• Lecture 4: Noetherian topological spaces. Irreducible components. Dimension. Regular functions on affine varieties.
• Lecture 5: Regular functions on affine open sets. Presheaves and sheaves. Ringed spaces. Morphisms of ringed spaces. Morphisms between affine varieties correspond to morphisms between coordinate rings. Isomorphisms and examples.
• Lecture 6: Abstract affine varieties. Basic open sets are affine. Prevarieties. Gluing. The affine line with double origin and the projective line. Varieties. Projective algebraic sets.
• Lecture 7: Projective Nullstellensatz. Cones. Projective algebraic sets and cones. Projective algebraic sets are prevarieties. Polynomial maps and morphisms.
• Lecture 8: Conics in P^2 are isomorphic to P^1 but the homogeneous coordinate rings are not. Examples of morphisms: Veronese embedding, Segre morphism. Showed that P^n x P^m is projective. Showed that P^n is a variety.
• Lecture 9: Main theorem of projective geometry. Complete varieties. Complete varieties have no non-constant regular functions. Complements of hypersurfaces in P^n are affine. A hypersurface and projective variety in P^n always intersect.
• Lecture 10: Dimension theory. Projection from a point. Dimension of P^n.
• Lecture 11: Dimension theory: intersection with hypersurfaces decreases the dimension by 1. Dimension can be computed on dense open sets. Dimension of affine varieties.
• Lecture 12: Theorem on the the dimension of fibers. Tangent spaces and smoothness.
• Lecture 13: Jacobi criterion. Examples of smooth varieties. Plane curve singularities. Tangent cone. Blowups can be used to resolve singularities.
• Lecture 14: Blowups (continued). Birational maps. Rational varieties. Examples. Elliptic curves are not rational.
• Lecture 15: The 27 lines on the cubic surface.
• Lecture 16: Affine schemes. The spectrum of a ring: the set, the Zariski topology, basic open sets. Quasicompactness. Examples.
• Lecture 17: Rings of fractions. Sheaves over the basis of a topology. The structure sheaf on the spectrum of a ring. Stalks of the structure sheaf.
• Lecture 18: Scheme theoretic intersections. Projective schemes. Locally ringed spaces. General schemes and morphisms.
• Lecture 19: Hilbert function. Examples. Hilbert polynomials. Hilbert polynomials of hypersurfaces in P^n. Degree-genus formula.
• Lecture 20: Bezout's theorem. Examples.
• Lecture 21: Applications of Bezout: Pascal's theorem, number of singularities of irreducible curves. Preview of Math 203b.

Homework