MATH 465 BASIC ALGEBRAIC GEOMETRY LECTURE 9
DAVID HYEON, SEPTEMBER 15 2003
4. Quasi-projective varieties Notation. For any polynomial F , we shall let F[j] denote the homogeneous degree j part of F , j = 0, . . . , deg F . 4.1 Closed subsets of projective
Math 465 Lecture Notes
Brendan Hassett January 30-February 11, 2004
4
4.1
Modules
Basic denitions
In the linear algebra of arbitrary rings, modules play the rle of vector spaces: o Denition 4.1 Let A be a ring. An A-module M is an additive abelian group M
Math 465 Lecture Notes
Brendan Hassett April 7-12, 2004
8
Cohomology on ane varieties
In this section, we consider algebraic varieties with the Zariski topology. The main reference is [Se].
8.1
Covering lemmas
Lemma 8.1 (Quasi-compactness lemma) Let X be
Math 465 Lecture Notes
Brendan Hassett April 14-19, 2004
9
9.1
Cohomology of projective varieties
Cohomology of twisting sheaves
Denition 9.1 Let F be a coherent OPn -module. For each integer d, the Serre twist F(d) is dened by the gluing data F(d)|Uj :=
Math 465 Lecture Notes
Brendan Hassett January 23, 2004
2
2.1
Zariski topology
Abstract topological spaces
Denition 2.1 A topology on a set X is a collection of subsets U X, called open subsets, with the following properties: 1. and X are open; 2. the int
Math 465 Lecture Notes
Brendan Hassett March 31-April 7, 2004
7
7.1
Cech cohomology
An obstruction computation
Let X be a topological space and consider an exact sequence of sheaves of abelian groups on X 0 K F G 0. Taking global sections gives an exact
LECTURE 6 MATH 465
2.2 Regular functions on a closed subset Denition. 1) A function f on a closed set X An is said to be regular if a polynomial F (x) k[x] such that F (p) = f (p), p X. 2) Regular functions with addition and multiplication form a ring k[X
Math 465 Lecture Notes
Brendan Hassett January 26-28, 2004
3
3.1
Localization
Fundamental open subsets as ane varieties
D(x1 ) = cfw_a1 A1 (k) : a1 = 0 A1 (k).
Example 3.1 Consider the fundamental open subset
We show how this can be given the structure of
Math 465 Lecture Notes
Brendan Hassett March 8-22, 2004
5
Sheaves
The seminal reference for this material is [Se].
5.1
Basic denitions
Throughout, X denotes a topological space. Denition 5.1 A presheaf P of groups on X assigns to each open U X a group P(U
Math 465 Lecture Notes
Brendan Hassett January 16, 2004
1
1.1
Introduction
First guiding problem: Interpolation
This should be compared to the discussion in chapter 12 of interpolation and Hilbert functions. We set up the notation. As usual, k[x1 , . . .
Math 465 Lecture Notes
Brendan Hassett March 24-29, 2004
6
Coherent sheaves
We continue to follow [Se].
6.1
Sheaves of rings and modules
Let X be a topological space. Denition 6.1 A sheaf of rings A on X is a sheaf with the following additional properties