Math 203A - Algebraic Geometry
Welcome to Math 203a!
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 will
cover affine and projective varieties corresponding roughly to the first
chapter of Hartshorne.
The course description can be found
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: MWF, 10:00-10:50, AP&M 7-421.
- Wednesday 1-2pm in AP&M 6-101.
Textbook: I will roughly follow
Andreas Gathamnn's notes available
I recommend that you also consult Shafarevich's Basic
Algebraic Geometry and Hartshorne's Algebraic
lecture or by appointment. Also, feel free to drop in if you see me in my
Other useful texts are
- Joe Harris, Algebraic Geometry: a first course.
- David Mumford, Algebraic Geometry I, Complex projective
- David Mumford, The red book of varieties and schemes
- 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
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 due in class.
- First class: Friday, September 29.
- Veterans Day: Friday, November 10.
- Thanksgiving break: November 23-24.
- Last class: Friday, December 8.
- Lecture 1: Introduction. Affine algebraic sets. Zariski
- Lecture 2: Ideals of affine algebraic sets. Weak and strong
Nullstellensatz. Correspondence between radical ideals and affine
- Lecture 3: More on irreducible sets. Irreducible components.
- Lecture 4: Coordinate rings. Fraction field. Regular
functions on affine
varieties. Basic open sets and their regular functions.
- Lecture 5: Presheaves and sheaves. Stalks. Ringed spaces.
- Lecture 6: Morphisms between affine varieties.
Corespondence with morphisms between coordinate rings.
- Lecture 7: More on morphisms. Products of varieties. Abstract
affine varieties. Rational maps, dominant maps, birational isomrphisms,
correspondence with fraction fields.
- Lecture 8: More on rational and birational maps and examples.
- Lecture 9: Morphisms of prevarieties. Products of
prevarieties. Varieties. Examples.
- Lecture 10: Projective space, Zariski topology, projective
algebraic sets, some examples.
- Lecture 11: Homogeneous coordinate rings, the sheaf of regular
functions on projective varieties, projective varieties are prevarieties,
morphisms and some examples.
- Lecture 12: Examples of morphisms: rational normal curves,
Veronese embedding, Segre morphisms.
- Lecture 13: Segre morphisms continued, and some consequences.
Main theorem of projective varieties.
- Lecture 14: Complete varieties. Regular functions on
complete varieties are constant.
- Lecture 15: Dimension theory for projective varieties.
Projection from a point. Comparing the dimension of a variety to that of
- Lecture 16: Dimension of projective space. Dimension of
intersections with hypersurfaces.
- Lecture 17: Dimension of varieties via open subsets. Dimension
as transcendence degree of the field of fractions.
- Lecture 18: Dimension of intersections with hypersurfaces.
- Lecture 19: Theorem on dimension of fibers. Smoothness.
Tangent space. Tangent cone. Examples of singularities.
- Lecture 20: Intrinsic definition of the tangent space. Affine
and projective Jacobi criterion.
- Lecture 21: Dimension of the singular locus. A brief
discussion of factorial varieties and normal varieties.
- Lecture 22: Codimension 1 phenomena on factorial varieties. A
brief discussion of normalization.
- Lecture 23: Blowup of the affine space at a point.
Exceptional hypersurface. Strict
- Lecture 24: Blowups in general and some properties.
- Lecture 25: Hilbert functions. Motivation and some
examples. First properties.
- Lecture 26: Hilbert polynomials. Encoding the dimension and
- Lecture 27: Bezout's theorem. Discussion of intersection
- Lecture 28: More on intersection multiplicities. Some
applications of Bezout's theorem. Number of singular points of irreducible
Homework 1 due Friday, October 6 - PDF
Homework 2 due Monday, October 16 - PDF
Homework 3 due Friday, October 20 - PDF
Homework 4 due Friday, October 27 - PDF
Homework 5 due Friday, November 3 - PDF
Homework 6 due Wednesday, November 15 - PDF
Homework 7 due Friday, December 1 - PDF
Homework 8 due Friday, December 8 - PDF