MATH 682-683
Algebraic Geometry, Winter-Spring 2012

Lectures:

MWF 2, Deady 206

There will be a make up lecture on May 31, at 10 am, in Deady 104

The following handout gives another way to prove Proposition 4.2.4 from the notes (the proof in the notes contains a mistake)

Office hours:

Tuesday 12-1, or by appointment

My office:

207 Deady, phone 3465635, e-mail apolish@uoregon.edu

Textbook:

During winter term we were using online notes by Andreas Gathmann, available here.
This term until further notice we will use "Algebraic geometry" by Hartshorne.

Pre-requisites:

Abstract algebra course such as Math 647-649. A course in commutative algebra would be helpful too. Otherwise, you will have to learn the necessary tools from commutative algebra in addition to the material from algebraic geometry - I could provide some guidance. My favorite books in commutative algebra are: 1) Atiyah, Macdonald; 2) Matsumura.

Homework

There will be no exams in this course, so the grade will be based solely on your homework scores. Homework should be turned in on the day when it's due either in class or at my office (you can slip it under the door). Homework must be stapled. You are encouraged to collaborate on homeworks, however, writing up the solutions should be an individual work.

Spring homework assignments:

Assignment #1 (due April 16): Hartshorne, ch. II, Exercises 2.1, 2.3, 2.4, 2.9, 2.10, 2.13abc, 3.6, 3.8 (for the last assertion you will need Thm. 3.9A from ch.I).

Assignment #2 (due April 27): Hartshorne, ch. II, Exercises 3.5, 3.7, 3.11cd, 3.17bcde, 3.18abc, 4.2, 4.4, 4.8, 4.9.

Assignment #3 (due May 14): Hartshorne, ch. II, Exercises 5.1, 5.6abc, 5.7, 5.16ab, 6.1, 6.4, 6.5ab.

Assignment #4 (due June 8): Hartshorne, ch. II, Exercises 7.1, 7.2, 7.3; ch. III, Exercises 3.7, 4.3, 4.7.

Rough guide for further study of Hartshorne's book

  • Finish II.7 and II.8, do Exercises 7.5, 7.6, 7.8-7.11, 8.1-8.6, 8.8
  • In the beginning of Ch.III study Theorems 2.7 and 3.7, Ex. 2.3, 3.1, 3.3-3.7, 4.1, 4.4, 4.5, 4.8.
  • Finish III.5, do Ex. 5.1-5.8
  • Study III.6-III.10, try to do as many exercises as possible
  • Chapters IV and V contain many beautiful applications of the general theory to curves and surfaces (we did some parts of ch.IV)

    • Winter homework assignments:

    Assignment #1 (due January 18): Exercises 1.4.1(ii), 1.4.4, 1.4.5(ii),(iii), 1.4.6. Also, the following two exercises:

  • Exercise A. Prove that every open cover of a Noetherian topological space admits a finite subcover.
  • Exercise B. Let p_1,...,p_k be all (distinct) irreducible factors of a polynomial f(x_1,...,x_n). Prove that Z(p_1),...,Z(p_k) are exactly the irreducible components of Z(f).

    • Assignment #2 (due February 3): Exercises 2.6.1, 2.6.2, 2.6.3, 2.6.4(iv), 2.6.5, 2.6.6.

    Assignment #3 (due February 17): Exercises 2.6.9, 2.6.10, 3.5.1, 3.5.2, 3.5.3, 3.5.4.

    Assignment #4 (due March 9): Exercises 3.5.7, 4.6.1, 4.6.2, 4.6.3, 4.6.7, 4.6.8.

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