Prerequisites: Math 250ABC, but please contact me if you are
interested and have not taken these courses
Textbook: Hartshorne, Algebraic Geometry
Syllabus: We will cover the basics of classical algebraic
geometry of affine and projective varieties defined by polynomial
equations. See also the
department syllabus.
Grading: 75% homework, 25% takehome final exam
Homework: Homework will be assigned roughly weekly
Welcome to Math 248A: Algebraic Geometry
Algebraic geometry is the study of solutions of systems of polynomials
equations. It is a classical field with a long history, which
has a close relationship to many fields of pure math, but has also recently
been applied to areas as diverse as engineering, computer
graphics, cryptography, and algebraic statistics, to name a few.
Because of the classical focus for 248A, I hope it will be accessible
to a broad audience, including applied math students.
Lecture notes
I shall, from time to time, post lecture notes on topics supplemental
to what is covered by Hartshorne. I see that a theme will soon emerge:
Hartshorne omits a discussion of abstract varieties, and likewise omits
a bridge from Chapter I to Chapter IV, so the notes will likely focus
on filling in these topics.
Recovering geometry from
categories: this is a purely optional note for those who are
categorically inclined, expanding on my remarks from class.
Varieties via atlases:
we describe how to define abstract varieties glued together from affine
varieties, using atlases as in classical differential geometry.
Nonsingular curves:
we study abstract nonsingular curves, showing that they are always
quasiprojective, and can always be compactified to nonsingular projective
curves. Includes a brief section on normality and normalization.
Chevalley's theorem and complete
varieties: we first prove Chevalley's theorem on images of morphisms.
We then define and explore the notion of completeness, analogous to
compactness of topological spaces. We prove that completeness may be
understood in terms of extending morphisms from nonsingular curves.
Problem sets
Problem sets will be posted here on Fridays, due the following Friday
in class. You are encouraged to collaborate with other students, as
long as you do not simply copy their answers.
Problem set #1, due 10/9: Do Exercises 1.1 (part (c) is
extra credit), 1.2, 1.3, 1.4, 1.8, 1.10, and 1.12 of Chapter I.
Selected solutions for problem set #1.
Problem set #2, due 10/16: Prove the following statements
directly from definitions:
(a) if X and Y are affine varieties in A^n and
A^m respectively, and f=(f_1,...,f_m) is a tuple of polynomials in n
variables such that f(X) is contained in Y, then f gives a morphism from
X to Y.
(b) If X is an affine variety, and f is a regular function on X,
then f is a morphism to A^1.
Also do Exercises 3.1 (a) (replace A^1-{0} by Z(xy-1) in A^2),
3.2, 3.3, 3.11 (assume X is affine), 3.15, and 3.19 (part (b) is extra
credit) of Chapter I.
Selected solutions for problem set #2.
Problem set #4, due 10/30: Do Exercise 3.14 of Chapter I,
and Exercises 2.8, 2.10, 2.11 and 3.1 of the Varieties via atlases
lecture notes.
Selected solutions for problem set #4.
Problem set #6, due 11/13: Do Exercises 3.2 and 3.3 of the
Nonsingular curves lecture notes, and Exercises 6.1, 6.2 and 6.7
of Chapter I.
Selected solutions for problem set #6.
Problem set #7, due 11/20: Do Exercises 4.4(c), 4.5, 5.10 and
6.6 of Chapter I.