Math 256A - Section 1 - Algebraic Geometry

Instructor: Brian Osserman

Lectures: TuTh 2:00-3:30pm, Room 103 Moffitt

Course Control Number: 55072

Office: 767 Evans, e-mail:

Office Hours: T 1:00-2:00, W 2:00-3:00

GSI: Shenghao Sun, 1056 Evans

GSI Office Hours: M 2:00-3:00, WF 11:00-12:00

Prerequisites: 250A; 250B strongly encouraged

Required Text: Hartshorne, Algebraic Geometry

Recommended Reading: Eisenbud and Harris, the Geometry of Schemes

Syllabus: This is the first semester of an integrated, year-long course in algebraic geometry. Although the primary source text will be Hartshorne's Algebraic Geometry, we will start from the beginning with schemes to emphasize their close connection with classical varieties and geometry, and we will supplement the text heavily with additional topics from both modern and classical algebraic geometry. The goal for the semester will be to cover enough that we can spend the last few weeks discussing deformation theory, and applications of scheme theory to classical problems. We will not assume any prior background in algebraic geometry.

Grading: 75% homework, 25% final exam.

Homework: Homework will be assigned roughly weekly

Final exam: Takehome exam, due on 12/11.


Welcome to Math 256A: Algebraic Geometry

Homework assignments and supplementary notes, as well as lecture synopses, will be posted here as the course progresses. In the mean time, here are some introductory comments.

My goals for this year-long course are:

This means we have a lot to accomplish. We will use Hartshorne's Algebraic Geometry as our primary source text, but we will start from the beginning with schemes to emphasize their close connection with classical varieties and geometry, and we will supplement the text heavily, with readings of a motivational nature, as well as with additional topics from both modern and classical algebraic geometry.

A note on difficulty: the course will be demanding, but hopefully also rewarding. If it's any consolation, I'm sure it will be just as difficult for me as for you. Algebraic geometry is uniquely situated at the intersection point of many fields: commutative algebra, differential geometry, algebraic topology, and number theory, to name some with the most obvious influences. While most definitions in the field have a clear intuitive picture behind them, it is often possible for the picture to get lost under a mountain of technicalities. To give an idea of the scope of the problem, Grothendieck's EGA, intended simply as a survey of the basic technical tools of the field, runs some 1500 pages — despite the fact that only four of an intended 13 chapters were ever written! My response to this dilemma is to focus as much as possible on bringing out the picture behind the definitions, and giving you a sense of the overall lay of the land in the field, while keeping the course accessible to those who have no prior experience with algebraic geometry. To this end, I will assign a lot of homework. I will also systematically cite proofs of statements from commutative algebra rather than doing them in lecture, and I will likely do the same with some results from algebraic geometry. In both cases, there are excellent references available for the proofs, so I feel that omitting them is the lesser of two (or several) evils.

Note on prerequisites: the main background material for this class will be commutative algebra. The amount of material required is fairly modest, but includes good familiarity with localization of rings, Noetherian rings, and basic theory of modules. Students are welcome to try to learn this material on their own; the book by Atiyah and MacDonald (specifically, chapters 1-3, and the first section of chapter 7) gives a compact account of the recommended material.


Problem sets

Problem sets will be posted here, due on Thursdays. Problems will be frequently assigned out of Hartshorne, so if you don't have your own copy, be ready to get problems from the library before it closes for the weekend. You may use any results from Hartshorne which have been stated in lecture, even if the proof was omitted.


Lectures

I will post here a synopsis of each lecture, including the sections in Hartshorne covered, and/or any supplemental notes.


Cheat sheets

This is a collection of cheat sheets, which are designed to serve two main purposes: first, to help organize all the definitions and basic results involving properties and schemes and morphisms, and second, to include statements and references for more general results when Hartshorne only states them under unnecessarily restrictive hypotheses.

These will be updated and expanded as we cover additional material.

Last updated: 12/7