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: WF 3:00  4:20 PM, AP&M 7321 .
Office hours:
 Wednesday 1:45  2:45 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, typically on Fridays.
Important dates:
 First class: Friday, September 27
 Veterans Day: Monday, November 11
 Thanksgiving break: November 2829
 Last class: Friday, December 6
Makeup lecture on Monday October 21 in 6402.
Lecture on Wednesday October 23 is in B402A.
Lecture Summaries
 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
varieties. Grassmannians
and Plucker embedding. Main theorem of elimination theory. Functions on
projective varieties
 Notes
 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.
Normalization.
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:
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