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 will
cover affine and projective varieties corresponding roughly to the first
chapter of Hartshorne.
The course description can be found
here.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: MWF, 10:0010:50, AP&M 7421.
Office hours:
 Wednesday 12pm in AP&M 6101.
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 Shafarevich's Basic
Algebraic Geometry and Hartshorne's Algebraic
Geometry.
Other useful texts are
 Joe Harris, Algebraic Geometry: a first course.
 David Mumford, Algebraic Geometry I, Complex projective
varieties
 David Mumford, The red book of varieties and schemes
Additional resources:
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.
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 due in class.
Important dates:
 First class: Friday, September 29.
 Veterans Day: Friday, November 10.
 Thanksgiving break: November 2324.
 Last class: Friday, December 8.
Lecture Summaries
 Lecture 1: Introduction. Affine algebraic sets. Zariski
topology.
 Lecture 2: Ideals of affine algebraic sets. Weak and strong
Nullstellensatz. Correspondence between radical ideals and affine
algebraic sets.
 Lecture 3: More on irreducible sets. Irreducible components.
Dimension.
 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.
Examples.
 Lecture 6: Morphisms between affine varieties.
Corespondence with morphisms between coordinate rings.
Isomorphisms. Examples.
 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.
Prevarieties. Glueing.
 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
the projection.
 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
transform.
 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
degree. Examples.
 Lecture 27: Bezout's theorem. Discussion of intersection
multiplicities.
 Lecture 28: More on intersection multiplicities. Some
applications of Bezout's theorem. Number of singular points of irreducible
plane curves.
Homework:
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