Positivity in Algebraic Geometry



We are reading Positivity in Algebraic Geometry I by Robert Lazarsfeld. The current plan is to cover Chapter 1 (Ample and Nef Bundles) and Chapter 2 (Linear Series) very thoroughly, and then use this foundation to cover some important consequences: the Lefschetz hyperplane theorem and the Kodaira vanishing theorem. If you’d like to be kept updated, please join the mailing list.


Date Speaker Reading
Week 1 (February 8) John Cobb 1.1 Preliminaries: Divisors, Line Bundles, and Linear Series (17 pages)
Week 2 (February 15) Alex Hof 1.2-3 The Classical Theory, \(\mathbb{Q}\)-Divisors and \(\mathbb{R}\)-Divisors (26 pages)
Week 3 (February 22) Colin 1.4 Nef Line Bundles and Divisors (20 pages)
March 4 (March 1) Lazarsfeld Pythagorean triples and parametrized curves
Week 5 (March 22) Asvin 1.5 Examples and Complements (18 pages)
Week 6 (March 29) Ivan 1.8 Castelnuovo-Mumford Regularity (21 pages)
Week 7 (April 5) Yifan 2.1 Asymptotic Theory (18 pages)
Week 8 (April 12) John Cobb 2.2 Big Line Bundles and Divisors (18 pages)
Week 9 (April 19) Daniel Erman (online!) 2.3 Examples and Complements (15 pages)
Week 10 (April 26) Colin 4.1-3 Kodaira and Nakano Vanishing Theorems (21 pages)

I’m leaving some room to split any readings into two weeks, or to veer in another direction if we would like. I know that Daniel Erman would like to give a lecture. By coincidence, Lazarsfeld will be in town around the middle of the semester.

Meeting Structure

We will be meeting weekly at 4:00-5:00 on Tuesdays in Van Vleck B231. While reading, write any questions you have and bring them. Each week the speaker will be giving a ~50 minute talk summarizing and motivating the reading that week, preferably picking a few of the many examples to work through in detail. Note that being the speaker does not mean you are expected to know everything – You can have lots of questions too. During the talk, you should ask your questions for everyone to collectively discuss, perhaps saving any “harder” questions for the end.