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: WF 3:00 - 4:20 PM, AP&M 7321 .
- Wednesday 1:45 - 2:45 in
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, typically on Fridays.
Make-up lecture on Monday October 21 in 6402.
Lecture on Wednesday October 23 is in B402A.
- First class: Friday, September 27
- Veterans Day: Monday, November 11
- Thanksgiving break: November 28-29
- Last class: Friday, December 6
- Lecture 1: Introduction. Affine algebraic sets. Zariski
topology. Ideals of affine algebraic sets. Nullstellensatz. Dictionary
between ideals and affine algebraic sets - Notes
- Lecture 2: Irreducibility. Noetherian topological spaces.
Irreducible components. Functions on affine varities. Coordinate rings.
Functions on open subsets - Notes
- Lecture 3: Regular functions on basic open sets.
Presheaves and sheaves. Stalks. Ringed spaces - Notes
- Lecture 4: Morphisms between affine varieties and morphisms
between coordinate rings. Rational maps, dominant maps, birational maps
and fraction fields. Examples - Notes
- 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 - Notes
- Lecture 6: Varieties. Projective space, projective algebraic
sets. Zariski topology - Notes
- Lecture 7: Regular functions on projective varities.
Projective varieties are prevarieties. Morphisms of
projective varieties. Examples. Rational normal curves. Veronese
embedding. Segre embedding - Notes
- Lecture 8: More on Segre embedding. Projective varieties are
and Plucker embedding. Main theorem of elimination theory. Functions on
- Lecture 9: Complete varieties. Morphisms from complete
varieties are closed. Projective varieties are complete. Dimension theory.
General discussion and goals. Projections from a point - Notes
- Lecture 10: Dimension theory for projective varieties.
Projection from a point. Comparing the dimension of a variety to that of
the projection. Dimension of projective space. Noether normalization - Notes
- Lecture 11: Finite maps and some properties. Geometric form of
Noether normalization. Intersection with hypersurfaces - Notes
- Lecture 12: Dimension of arbitrary varieties. Dimension via
the transcendence degree of the field of rational functions. Dimension of
affine intersections with hypersurfaces - Notes
- Lecture 13: Theorem of dimension of fibers. Smoothness.
Zariski tangent space - Notes
- Lecture 14: Intrinsic nature of the Zariski tangent space.
Tangent cone. Smooth points. Jacobi criterion. Semicontinuity. Singular
set is closed - Notes
- Lecture 15: Every variety is birational to a hypersurface.
Smooth locus is nonempty. Tangent spaces and morphisms. Tangent spaces of
projective space. Bertini's theorem - Notes
- Lecture 16: Factorial varieties. Normal varieties.
Motivation for studying blowups - Notes
- Lecture 17: Blowups. Blowup of the plane at the origin. Strict
transform. Exceptional set. Resolving singularities of plane curves
General construction. del Pezzo surfaces - Notes
- Lecture 18: More on blowups. Invariants of projective
varieties. Questions and motivation. Hilbert function. Examples. First
properties - Notes
- Lecture 19: Hilbert polynomial. Reading dimension, degree and
arithmetic genus from the Hilbert polynomial. Examples. Global Bezout's
theorem - Notes
- Lecture 20: Local Bezout's
theorem and multiplicities. Number of singularities of plane curves.
Review - Notes
Homework 1 due Friday, October 4 - PDF
Homework 2 due Friday, October 18 - PDF
Homework 3 due Friday, October 25 - PDF
Homework 4 due Friday, November 1 - PDF
Homework 5 due Friday, November 15 - PDF
Homework 6 due Friday, November 22 - PDF
Homework 7 due Friday, December 6 - PDF