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 three 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 course description can be found here .

Kiran Kedlaya will teach Math 203b and Math 203c in the Winter and Spring.

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

Lectures: WF, 11-12:20, AP&M 7-421.

Office hours:

Wednesday 1-2 PM 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.

Prerequisites:

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.

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.

Important dates:

• First Class: September 28.
• Veterans Day: November 12.
• Thanksgiving break: November 22-23.
• Last day of classes: December 7.

Announcements:

• No Lecture on Friday Oct 12. There'll be a make-up lecture on Oct 15.
• No office hour on Wednesday Oct 10. I will make up for it on Monday Oct 15 from 1:30-2:30. I am also available for questions by email or on Wednesday immediately after class.
• No Lecture on Friday, November 16. There will be a make up lecture on Monday, November 19.

Lecture Summaries

• Lecture 1: Introduction. Affine algebraic sets. Zariski topology. Correspondence between ideals and affine algebraic sets. The weak and strong Nullstellensatz.
• Lecture 2: Irreducible topological spaces. Notherian topological spaces. Irreducible components. Dimension.
• Lecture 3: Regular functions on affine varities and on open subsets. Description of regular functions on basic open sets. Presheaves and sheaves. Stalks.
• Lecture 4: Ringed spaces. Morphisms of ringed spaces. Morphisms between affine varieties as ringed spaces and morphisms between coordinate rings. Rational maps, dominant maps, birational maps.
• Lecture 5: Abstract affine varieties. Basic open sets are affine. Prevarieties. Gluing prevarieties. Examples: projective line, the affine line with double origin. Products of prevarieties. Varieties and some examples.
• Lecture 6: Projective space, projective algebraic sets. Zariski topology. Regular functions on projective varities. Projective varieties are prevarieties.
• Lecture 7: Morphisms of projective varieties. Examples. Rational normal curves. Veronese embedding. Segre embedding. Products of projective varieties are projective.
• Lecture 8: Morphisms of projective varieties are closed. Complete varieties. Regular functions on complete varieties are constant.
• Lecture 9: Dimension theory for projective varieties. Projection from a point. Comparing the dimension of a variety to that of the projection. Dimension of projective space.
• Lecture 10: Dimension of arbitrary varieties. Theorem of dimension of fibers. Dimension of intersections.
• Lecture 11: Tangent space and tangent cone. Smooth and singular points. Examples. Ordinary r-fold points. Jacobi criterion.
• Lecture 12: IFT fails for the Zariski topology. Size of singular locus. Normal varieties and normalization. Examples.
• Lecture 13: Blowup of A^2 at the origin. Resolving singularities of plane curves. The exceptional hypersurface, the strict transform. Connection with the tangent cone. Blowups in general. Examples.
• Lecture 14: The 27 lines on a smooth cubic surface. Rationality of the smooth cubic surface.
• Lecture 15: Affine schemes: the prime spectrum, the Zariski topology. Examples. Affine varieties vs. affine schemes.
• Lecture 16: Rings of fractions. Sheaves over the basis of a topology. The structure sheaf over the spectrum of a ring. Stalks of the structure sheaf.
• Lecture 17: Locally ringed spaces and morphisms of locally ringed spaces. Fibered products (of affine schemes). Scheme theoretic fibers and scheme theoretic intersections. Arbitrary schemes. Projective schemes.
• Lecture 18: Hilbert polynomials. Degree. Arithmetic genus. Examples. Degree genus formula.
• Lecture 19: Bezout's theorem. Applications: automorphisms of P^n, Pascal's theorem, the number of singularities of degree d plane curves.
• Lecture 20: Divisors on smooth curves. Examples: projective line, the smooth plane cubic. Invertible sheaves. Correspondence between divisors and invertible sheaves.

Homework:

Homework 1 due Friday, Oct 5 PDF

• Solutions: PDF

Homework 2 due Monday, Oct 15 PDF

• Solutions: PDF

Homework 3 due Wednesday, Oct 24 PDF

Homework 4 due Friday, November 2 PDF

• Solutions: PDF

Homework 5 due Friday, November 9 PDF

• Solutions: PDF

Homework 6 due Monday, November 19 PDF

• Solutions: PDF

Homework 7 due Wednesday, November 28 PDF

• Solutions: PDF

Homework 8 due Friday, December 7 PDF

• Solutions: PDF