Math 273, Fall 2005: Introduction to Algebraic Geometry

Instructor: Chad Schoen

Prerequisites: Math 252 (Commutative Algebra) and Math 272 (Riemann Surfaces)
It may be possible to take math 272 concurrently. Math 272 supplies useful intuition.
It also supplies technical knowledge about sheaves and their cohomology. Math 252 is the more important
prerequisite.

The subject:

Algebraic geometry is the study of the solution sets of systems of polynomial equations. These objects are called algebraic varieties.

Course Goals:

The basic objects of study in a first course on algebraic geometry are quasi-projective varieties over algebraically closed fields. We will investigate basic properties of these objects and learn some of the tools applied to their study. Topics: Quasi-projective varieties with numerous examples, regular maps, rational maps, elimination theory, singularities, divisors and maps to projective space. To the extent that time permits we will treat coherent sheaves and their cohomology, cohomological invariants, rudiments of intersection theory, a peek at algebraic surfaces.

The course treats concepts which are essential to further work in algebraic geometry. To the extent that other areas of mathematics require understanding of basic algebraic geometry concepts, this course is important for students with a wide range of research interests. Most researchers in the following fields need to be familiar with the rudiments of algebraic geometry: number theory, algebra, algebraic groups, quadratic forms, singularities, complex analytic geometry, string theory, and perhaps algebraic topology and differential geometry.

Research in algebraic geometry requires an understanding of algebraic varieties over non-algebraically-closed fields and, more generally, of schemes. Although these topics will not be treated in this course, the course will prepare students to study these concepts.

Text

Algebraic Geometry, by Robin Hartshorne

Homework

For Monday September 5:
1. Read Hartshorne, p. 1-5.
(a) Show that a polynomial map between affine varieties is continuous in the Zariski topology.
(b) Let R be a unique factorization domain. Describe which principal ideals
in R are radical ideals. (Note: For a general ideal in a ring it is frequently
somewhat of a chore to determine if it is a radical ideal or not.)
(c) Find the irreducible components of the following affine varieties:
(i) V((x^2-y^2)y)
(ii) V((xy,xz))
(iii) V((xy,xz,yz))
(d) Work the following problems in part 1 of chapter I: 1a,b; 2; 3; 4; 5; 6; 7.
Work exercise set 1.

For Tuesday September 13:
Read Hartshorne, section I.2.
Work problems 2.2, 2.3, 2.4, 2.5, 2.9(b), 2.15, 2.16 in Hartshorne.
Handout on "Why projective space?": Look at problems 1,2, 5.
Handout on Grassmannians: Look at problems 1-9.

For Tuesday September 20:
Read Hartshorne, section I.3.
Look at problems 3.1, 3.2, 3.3, 3.6, 3.10, 3,14.
Handout on Grassmannians: Look at problems 10-14.
Look at problem 3 on the handout on projective space if you wish.
It is somewhat tangential to what we are working on.

For Tuesday September 27:
Read Hartshorne, section I.4.
Work problems 4.1-4.8.
Handout on Grassmannians: Look at problems 15-17.
Work problem 5.1 if you have extra time.

For Tuesday October 4:
Read Hartshorne, pages 31-33.
Work problems 5.1-5.6.
Complete the remaining problems on the handout on Grassmannians.

For Thursday October 13:
Work problems 3.7, 3.1(d), 3.15(a) and (d), 5.7, 5.8, 5.9, 5.11, 5.15.

For Tuesday October 18:
Read Hartshorne, page 27. (Note that all field extensions in
characteristic 0 are trivially separately generated.)
Read Hartshorne, pages 47-52.
Work problems 2.11, 2.12, 2.13, 3.4, 3.5, 7.1.

For Tuesday October 25:
Read Hartshorne, pages 53-54.
Work problems 7.2a,b , 7.3, 7.4, 7.5
Read Hartshorne, section I.8.

As indicated in class, Hartshorne introduces the language of schemes in the first part of chapter 2. My philosophy, and that of many others, is that the power of schemes are best appreciated after a one semester introduction to the classical theory of algebraic varieties. Thus we will skip over sections 2.2-2.4 in Hartshorne. This means that as you read later sections, you will be reading about schemes while thinking in terms of varieties. This should not be an overwhelming burden. The schemes associated to the varieties which we study are really quite similar to these varieties. One real advantage of schemes stems from the fact that they are very much more general than algebraic varieties. For example they are ideal for studying deformations of algebraic varieties. Since we won't be studying deformations, we will get by rather well without schemes. Reading the first paragraph or two in section 2.2 may help with the orientation to the new language. Read more of 2.2 if you feel the need to. You may have seen the basic definition of an affine scheme in your commutative algebra course. If so, it may provide a bit of help with the reading, but it should not be crucial. What Hartshorne says in the sections we will cover by and large makes fine sense for varieties.

For Tuesday November 1
Read Hartshorne, section 2.1(review of sheaves)
Read the first paragraph in section 2.2 for orientation
on the language of schemes. We will continue to work with
varieties over an algebraically closed field and you will
need to be able to translate from Hartshorne's language
to the classical language.
Read Hartshorne, pages 129-136.(Weil divisors)
Work problems in section I.7: 7.2c,d,e, 7.6,7.7,7.8.
Work problems in section II.6: 6.1.

For Tuesday November 8
Read Hartshorne, pages 140-146, (Cartier divisors and invertible sheaves)
Work problems in section II.6: 6.2, 6.3a,b , 6.4, 6.5, 6.8a, 6.9

For Tuesday November 15
Read Hartshorne, pages 149-151.(Morphisms to P^n)
Work problems in section II.7: 7.2, 7.3
Read Hartshorne, pages 108-118. (Sheaves of modules)
Work problems in section II.5: 5.5, 5.7
Handout on morphisms associated to linear systems:
Look at part I.

For Tuesday November 22
Read Hartshorne, III.1 and pages 206-208.
Work problems in section II.1: 1.8, 1.16, 1.21
Work problems in section III.2: 2.2
Handout on morphisms associated to linear systems parts I-III.

For Tuesday November 29
Read Hartshorne, III.4 and Lemma III.2.10.
Work problems in section II.5: 1.
Morphisms associated to linear systems parts IV-VI.
First exercises in cohomology: 1.

For Tuesday December 6
Read Hartshorne, III.5 and pages 156-159.
First exercises in cohomology: 2-5.