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 will post lecture notes here on topics supplemental to what is covered
by Hartshorne, most likely focused on abstract varieties and on providing
a bridge from Chapter I to Chapter IV.

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.

Projective varieties:
A development of projective varieties as examples of abstract varieties,
including a study of regular functions on projective varieties and
morphisms to projective varieties. Also, a brief discussion of completeness.

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.

Divisors on nonsingular curves:
We introduce the concept of divisors on nonsingular curves, and study
their properties and their relationship to morphisms to projective space.

Differential forms:
We introduce the concept of differential forms on nonsingular varieties,
and then specializing to the case of curves, look at the associated
divisors, as well as the relationship between differential forms and
ramification of morphisms.

The Riemann-Roch and Riemann-Hurwitz
theorems:
We state the Riemann-Roch theorem without proof, and give a brief
sketch of some applications, including a proof of the Riemann-Hurwitz
theorem.

Problem sets

Problem sets will be posted here on Monday, due the following Monday
in class. You are encouraged to collaborate with other students, as
long as you do not simply copy their answers. Beyond this, you should
not seek any external help.

Problem set #1, due 10/7: Do Exercises 1.1 (part (c) is
extra credit), 1.2, 1.3 and 1.4 of Chapter I of Hartshorne.
Selected solutions for problem set #1.

Problem set #4, due 10/28: Do Exercises 3.15, 5.1, 5.2, 5.3,
5.4 (a) and (b), and 5.10 (b) and (c) of Chapter I of Hartshorne. For
3.15 (c), interpret "variety" to mean "quasiaffine variety".
Selected solutions for problem set #4.

Problem set #5, due 11/4: Do Exercises 2.2, 2.5, 2.9 and 2.10
of the Abstract varieties via atlases lecture notes. Also:

Prove or disprove and salvage if possible: a nonsingular affine
algebraic set is irreducible.

The only regular functions on the entire projective line (as defined in
Example 1.7 of the lecture notes) are constant.

Problem set #6, due 11/13: Do Exercise 3.1
of the Abstract varieties via atlases lecture notes, and Exercises
2.17 (a)-(b), 3.4 and 3.5 of Chapter I of Hartshorne. Also:

Show that any prevariety X has a Noetherian topological space.

If X is a prevariety and U an open subprevariety, then dim U = dim X.
If Z is a closed subprevariety, then dim Z + codim_{X} Z = dim X.

Show that if X is a prevariety and Y is a variety, then two
morphisms from X to Y which agree on a nonempty open subset of X must
be the same.

Problem set #8, due 11/25: Do Exercise 2.2 of the
Nonsingular curves lecture notes, and Exercises 6.1, 6.2, 6.6(c)
and 6.7 of Chapter I of Hartshorne.
Selected solutions for problem set #8.

Problem set #9, due 12/2: Do Exercise 6.3 of Chapter I of
Hartshorne, and also show that a morphism from the projective line to
itself is ramified at all points if and only if it is ramified at
infinitely many points, if and only if the characteristic is p and the
morphism factors through the Frobenius (i.e., pth power) map.
Selected solutions for problem set #9.

Final Exam

There will be a takehome final exam for the course.

The takehome final exam is due Wednesday, December 11 by 5:00. Obviously,
you are not allowed to discuss it with any of your classmates or anyone
else until after you have handed it in. You may email me with questions
or concerns, but I will not give out any hints individually; if I
determine that a hint is appropriate, I will email it to everyone.