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 (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
will teach Math 203b and Math 203c in the Winter and Spring.
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: WF, 11-12:20, AP&M 7-421.
- Wednesday 1-2 PM in AP&M 6-101.
Textbook: There is no required textbook. I will follow
Andreas Gathamnn's notes available online
lecture or by appointment. Also, feel free to drop in if you see me in my
Other useful texts are
- Igor Shafarevich, Basic Algebraic Geometry I, Varieties in
- Joe Harris, Algebraic Geometry: a first course.
- David Mumford, Algebraic Geometry I, Complex Projective
More advanced but useful references are:
- Robin Hartshorne, Algebraic
- 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.
- First Class: September 28.
- Veterans Day: November 12.
- Thanksgiving break: November 22-23.
- Last day of classes: December 7.
- No Lecture on Friday Oct 12. There'll be a make-up lecture on Oct
- 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 1: Introduction. Affine algebraic sets. Zariski
topology. Correspondence between ideals and affine algebraic sets. The
weak and strong
- 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
- 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
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
- 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
- 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
- Lecture 20: Divisors on smooth curves. Examples: projective
line, the smooth plane cubic. Invertible sheaves.
Correspondence between divisors and invertible sheaves.
Homework 1 due Friday, Oct 5 PDF
Homework 2 due Monday, Oct 15 PDF
Homework 3 due Wednesday, Oct 24 PDF
Homework 4 due Friday, November 2 PDF
Homework 5 due Friday, November 9 PDF
Homework 6 due Monday, November 19 PDF
Homework 7 due Wednesday, November 28 PDF
Homework 8 due Friday, December 7 PDF