# Math 245A Topics in algebraic geometry: Introduction
to intersection theory in algebraic geometry

**Lectures:** Mondays 9-10:50 and Wednesdays 10-10:50 (not the
times listed in the course guide) as well as Friday Oct. 1, 10-10:50.
**Office hours:** By appointment, in 380-383M (third floor of the
math building). I'll usually be in Mondays and Wednesdays 2:15-3 (my
210A office hours). I will almost always be available to talk at
length after each class, and at other times of the week as well.

**Our goal:** We'll develop intersection theory in
algebro-geometric context, which will allow us to deal with singular
spaces. In *Intersection theory*, Fulton develops a very
powerful machine quite cleanly and quickly. He requires a reasonable
comfort level with schemes. If you haven't seen them before and are
ambitious, you should be able to follow the class. (This will involve
chatting with me about any points you find confusing.) The technical
tools are developed in the first eight or so chapters. The remaining
chapters are important applications, and can be read in almost any
order once the basic theory is developed. I would like to cover the
first 8 chapters (including Chern classes, and intersection theory on
nonsingular varieties), and then (in some order) Chapters 9 (excess
and residual intersection), 14 (degeneracy loci and Grassmannians),
and 15 (Riemann-Roch for nonsingular varieties); most likely we'll settle
for less.

**Reference:** Fulton's
*Intersection theory* (either edition). It will be very helpful if
you have a copy.

**Homework:** Unlike most advanced graduate courses, there likely
will be homework. If there is any, it won't be onerous; your main task
will be to understand the subject.

**Notes:**
Notes for many of the classes in ps and pdf formats will be posted
here.
They are very rough, some rougher than others! For the most part
I will be following Fulton (with some additional explanation), and
I make no claim of originality.
(If the pdf file looks funny on your computer or printer, please
let me know, so I can try to fix it!) Thanks to Justin Walker
for many improvements.

Class 1 (Mon. Sept. 27): welcome, examples, strategy
(ps,
pdf).
Class 2 (Wed. Sept. 29): generic points, orders of rational functions
along Weil divisors, rational equivalence, Chow group, proper/projective/finite
morphisms
(ps,
pdf).
Class 3 (Fri. Oct. 1): proper/projective/finite
morphisms, proper pushforwards of cycles
(ps,
pdf).
Class 4 (Mon. Oct. 4): proper pushforward of rational equivalence,
flat morphisms, fundamental cycle, pulling back cycles and rational
equivalences, proper pushforward commutes with flat pullback
(ps,
pdf). (Homework assigned!)
Class 5 (Wed. Oct. 6, taught by Rob Easton): excision
exact sequence, affine bundles, Cartier divisors, pseudodivisors
and associated terminology.
Class 6 (Mon. Oct. 11):
more on divisors, intersecting with pseudodivisors, first Chern
class of a line bundle, Gysin pullback to a(n effective Cartier) divisor.
Homework "due".
(ps,
pdf).
Class 7 (Wed. Oct. 13):
proving various consequences of key theorem of Chapter 2; crash course
on blowing up
(ps,
pdf).
Class 8 (Mon. Oct. 18):
finishing Chapter 2; projective bundles and O(1), Segre classes,
properties of Segre classes
(ps,
pdf).
Class 9 (Wed. Oct. 20):
Chern classes and their properties
(ps,
pdf).
Class 10 (Mon. Oct. 25): the splitting principle; applications
of Chern classes
(ps,
pdf).
Class 11 (Wed. Oct. 27): Chow groups of
vector bundles and projective bundles.
(ps,
pdf).
Class 12 (Mon. Nov. 1): cones; Segre classes of cones and
subschemes.
(ps,
pdf).
Class 13 (Wed. Nov. 3): the Segre class of a subscheme
behaves with respect to proper pushforwards and flat pullbacks.
(ps,
pdf).
Class 14 (Mon. Nov. 8): multiplicity of a variety along a subvariety,
deformation to the normal cone, specialization to the normal cone, gysin pullback for lci's, intersection products on smooth varieties.
(ps,
pdf).
Class 15 (Wed. Nov. 10, taught by Andy Schultz): linear series.
(ps,
pdf).
Class 16 (Mon. Nov. 15): intersection products, refined Gysin homomorphisms, i shriek.
(ps,
pdf).
Class 17 (Wed. Nov. 17): i shriek again; refined Gysin pullback
behaves well (commutes with proper pushforward and flat pullback and other
refined Gysins),
excess intersection formula and self-intersection, functoriality,
local complete intersection morphisms.
(ps,
pdf).
Class 18 (Mon. Nov. 22):
Grothendieck K groups,
Grothendieck-Riemann-Roch, proof for projective space.
(ps,
pdf).
Class 19 (Wed. Nov. 24):
Grothendieck-Riemann Roch proof for projective morphisms
between smooth varieties.
(ps,
pdf).
Class 20 (Mon. Nov. 29): Bivariant intersection theory
(ps,
pdf).
Class 21 (Wed. Dec. 1): Bivariant intersection
theory continued
(notes included in previous day's notes).

Back to my home page.

Ravi Vakil

Department of Mathematics Rm. 383M

Stanford University

Stanford, CA

Phone: 650-723-7850 (but e-mail is better)

Fax: 650-725-4066

vakil@math.stanford.edu