Math 818, Algebra and Geometry, Fall 2010

Math 818 is a graduate beginning algebraic geometry course. Lectures are 10:30 - 12:20, Tuesdays and Thursdays in AQ 5020.

Announcements

• If you would like to know more about abelian varieties in a language which is more sophisticated than what we've been using, but not hopelessly so, please look at Milne's course notes.
• The written project is due Dec 7 (in my mailbox).
• The schedule for the presentations is:

  Nov 30

  Jemma; Julian

  Dec 2

  Tom; Sam; James

• Office hours are Mondays 10:30-11:30 and 2:30-3:30. You should also feel free to drop by whenever convenient.
• We will meet Thursday September 9 to decide amongst us what topics to cover and how you will be evaluated. Regular lectures will begin Tuesday September 14.

Assignments

1. Due Thursday Oct 7 in class. Solutions.
   • Shafarevich I.2.5, I.2.11, I.2.15.
   • Fulton 1-10, 1-26.
   • Shafarevich I.3.4, I.3.6.
   • Draw a diagram illustrating the relationships between the algebraic set picture, the ideal picture, coordinate ring picture, and the function field picture in the affine case. Try to include as many relationships as we've discussed and to make it a clear and useful big picture view of what we've learned so far.
   • Fulton 4-18, 4-20 (see chapter 2 section 6 for the star notation, which is not the same as in question 4-18).
2. Due Thursday Oct 28 in class. Solutions.
   • Shafarevich I.4.9
   • Shafarevich I.5.7, I.5.8
   • Fulton 3-2, 3-8, 3-13
   • Fulton 3-20
   • Fulton 5-3, 5-6, 5-22
3. Due Thursday November 18. Solutions.
   • Shafarevich I.6.6, I.6.7, I.6.8, II.1.1
   • Fulton 7-2. F' means blow up the singularity.
   • Shafarevich II.4.4
   • Shafarevich III.1.2, III.1.5
   • Shafarevich III.1.12, III.1.18
4. Due Thursday December 2. Solutions.
   • Consider Fulton's and Shafarevich' proofs of Bezout's theorem for curves (Fulton 5.3, Shafarevich III.2.2). Are they fundamentally the same proof in different languages or do they have really different ideas? If they are different summarize the key ideas of each in a few lines. If they are the same provide a dictionary to translate the language and notation between them.
   • Fulton 8-10, 8-14 parts a and b.
   • Shafarevich III.3.1
   And that's all.

Information

Outline (in pdf format).

Instructor:

Karen Yeats

Office:

SC K 10508

Email:

karen_yeats at sfu.ca

Office Hours:

Mondays 10:30-11:30 and 2:30-3:30.

Lectures:

10:30-12:20 Tuesdays and Thursdays AQ 5020

Textbooks:

Shafarevich Basic Algebraic Geometry 1 and Fulton Algebraic Curves.