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.
Kiran Kedlaya
will teach Math 203b and Math 203c in Winter and Spring.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: WF, 2:003:20, AP&M 7421.
Office hours:
 Wednesday 3:304:30 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 follow
Andreas Gathamnn's notes available online.
I recommend that you also own a copy of 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
 Igor Shafarevich, Basic Algebraic Geometry I, Varieties in
projective
space.
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 25.
 Veterans Day: Wednesday, November 11.
 Thanksgiving break: November 2627.
 Last class: December 4.
Announcements:

There will be no lectures on Oct 28 and Oct 30. Makeup
lectures will be scheduled on
Monday, October 12 and Monday, October 19, at the usual time.
 There will be no lecture on Wednesday, Nov 25, the day before
Thanksgiving. The makeup lecture is scheduled for Monday, Nov 23.
Lecture Summaries
 Lecture 1: Affine algebraic sets. Zariski topology.
Correspondence between ideals and affine algebraic sets. The weak and
strong Nullstellensatz. Irreducible topological spaces. Noetherian
topological spaces.
 Lecture 2: Irreducible components.
Dimension. Regular functions on affine varities and on open subsets.
 Lecture 3: Regular functions on basic open sets. Presheaves
and sheaves. Stalks. Ringed spaces. Morphisms of ringed spaces.
 Lecture 4: Morphisms between affine varieties and morphisms
between coordinate rings. Rational maps, dominant maps, birational maps
and fraction fields. Examples.
 Lecture 5: Abstract affine
varieties. Basic open sets are abstract affine varieties. Prevarieties.
Gluing prevarieties. Examples: projective line, the affine line with
double origin. Products of prevarieties. Varieties.
 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 hypersurface sections. Dimension of arbitrary
varieties. Dimension of intersections in affine or projective space.
 Lecture 11: Dimension via the transcendence degree of the
field of rational functions. Agreement with the old definition. Theorem on
dimension of fibers.
 Lecture 12: Dimension of hyperplane
sections final form. Tangent spaces and tangent cones. Smooth and singular
points. Examples. Ordinary mfold points.
 Lecture 13:
Intrinsic definition of tangent spaces. Affine and projective Jacobi
criterion. All varieties are birational to hypersurfaces. Existence of
smooth points on varieties.
 Lecture 14: Smoothness in algebraic and complex geometry  failure of the implicit function theorem.
Consequences of smoothness and codimension 1 phenomena. Normal
varieties.
 Lecture 15: Motivation for studying blowups. Blowup of the
plane at the origin. Resolving singularities of plane curves. The
exceptional hypersurface, the strict transform.
 Lecture 16: Blowups continued. Hilbert functions. Motivation
and examples. First
properties of Hilbert functions.
 Lecture 17: Existence of Hilbert polynomials. Reading off
the dimension and degrees of
projective varieties from Hilbert polynomials. Arithmetic genus.
 Lecture 18: Bezout's theorem  global and local form.
Intersection multiplicities.
 Lecture 19: Applications of Bezout's theorem. Number of
singularities of irreducible plane
curves. Divisors on curves. Degree of principal divisors. Smooth plane
cubics.
Homework:
Homework 1 due
Friday, October 2  PDF
Homework 2 due
Friday, October 9  PDF
Homework 3 due
Monday, October 19  PDF
Homework 4 due
Wednesday, November 4  PDF
Homework 5 due
Friday, November 13  PDF
Homework 6 due
Monday, November 23  PDF
Homework 7 due
Friday, December 4  PDF