MATH 203B, Winter 2021

Professor

Elham Izadi ; AP&M 6240 ; phone: 858-534-2638 ; email: eizadi@ucsd.edu ; Office hours: Mondays 14:00-15:00 ; Homework discussion sessions: Thursdays 13:30-15:00
Lectures: Tueday, Thursday 11:00-12:20

Course description

This is the second quarter of the introductory sequence in Algebraic Geometry. We will cover some of Chapters 2, 3 and 4 of Hartshorne's book, as time permits.

Prerequisites

MATH 203A

Text

Algebraic Geometry, By Robin Hartshorne

References

The litterature on Algebraic Geometry is staggering. Here are a few books but you can find many more with some basic google searches.

An invitation to algebraic geometry, by Karen Smith, Lauri Kahanpää, Pekka Kekäläinen, William Traves.
The red book of varieties and schemes, by David Mumford.
Principles of algebraic geometry, by Phillip Griffiths and Joseph Harris.
Complex algebraic surfaces, by Arnaud Beauville.
Algebraic geometry, a first course, by Joseph Harris.
Geometry of Schemes, by David Eisenbud and Joseph Harris.
Algebraic Geometry, An Introduction, by Daniel Perrin.
Algebraic geometry, by Igor Shafarevich.
Introduction to Algebraic Geometry, by Brendan Hassett.
Introduction to commutative algebra, by Michael Atiyah and Ian McDonald.
Commutative algebra, By H. Matsumura.
Commutative Algebra with a view toward algebraic geometry, by David Eisenbud.

Class Notes

01/05/2021 01/07/2021 01/12/2021 01/14/2021 01/19/2021 01/21/2021 01/26/2021 01/28/2021 02/02/2021 02/09/2021 02/10/2021 02/11/2021 02/16/2021 02/18/2021 02/23/2021 02/25/2021 03/02/2021 03/04/2021 03/09/2021 03/11/2021

Homework

There will be weekly homework assignments (posted below, but subject to change during the quarter). Please submit your homework online to Canvas as a pdf file by the due time and date. Late homework will not be accepted. Homework will normally be due one week after it is assigned, usually on a Tuesday. On the due date of a homework assignment, Canvas will automatically assign two of you as reviewers for each homework. The reviewers will write comments for the homework that they read. You will have two days, until the Thursday homework discussion session, to read and comment on the homework that is assigned to you as reviewer. At the Thursday homework discussion session, we will go over some of the solutions to the homework.

Homework assigments

Homework 1: Due Tuesday January 12
Chapter 2, Section 2 #14,16,17,18,19

Homework 2: Due Tuesday January 19
Chapter 2, Section 1 #18, Section 5 #1,3,5, and the following
(a) Given a locally free sheaf of finite rank F on a locally ringed space X, describe the transition matrices of the dual sheaf in terms of those of F.
(b) Describe the transition matrices of the tensor product of two locally free sheaves of finite rank F anf G in terms of those of F and G.

Homework 3: Due Tuesday January 26
Chapter 2, Section 5 #2,4,7,8,9

Homework 4: Due Tuesday February 2
Chapter 2, Section 5 #10,11,12,13

Homework 5: Due Tuesday February 9
Chapter 2, Section 5 #14,15,16,17,18 (note: these problems are especially hard but also especially important and interesting, do as much of them as you can, only please make sure you read them carefully and understand the statements, in many textbooks these are done in the text)

Homework 6: Due Tuesday February 16
Chapter 2, Section 6 #1,8, Section 7 #1
Also prove the following: If X is integral, then any nonzero morphism of invertible sheaves is injective, any generically injective morphism of locally free sheaves is injective (hint: first prove that a locally free sheaf has no torsion subsheaf, where by a torsion sheaf we mean a sheaf whose support has codimension > 0).

Homework 7: Due Tuesday February 23
Chapter 2, Section 6 #5,10, Section 7 #2,3

Homework 8: Due Tuesday March 2
Chapter 2, Section 7 #4,5,7,8,9

Homework 9: Due Tuesday March 9
Chapter 2, Section 8 #1,2,3

Homework 10: Due Tuesday March 16
Chapter 2, Section 8 #4, 5


Elham Izadi
Last modified: Tue Mar 9 15:22:30 PST 2021