## 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:

• To provide an accessible introduction to the modern language of algebraic geometry.
• To provide a good working knowledge of the most basic technical tools of algebraic geometry.
• To discuss the geometric intuition behind schemes.
• To give a glimpse of important technical topics beyond the scope of the course.
• To give a glimpse of some classical, recent, and current research areas.

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.

• Problem set #1, due 9/7: PS, PDF.
• Problem set #2, due 9/14: PS, PDF. If you did not complete exercises 6 and 7 on the first problem set, you may hand them in with this one.
• Problem set #3, due 9/21: PS, PDF.
• Problem set #4, due 9/28: PS, PDF.
• Problem set #5, due 10/05: PS, PDF.
• Problem set #6, due 10/12: PS, PDF.
• Problem set #7, due 10/19: PS, PDF.
• Problem set #8, due 10/26: PS, PDF.
• Problem set #9, due 11/2: PS, PDF.
• Problem set #10, due 11/9: PS, PDF.
• Problem set #11, due 11/16: PS, PDF.
• Problem set #12, part 1, due 11/30: PS, PDF.
• Problem set #12, part 2, due 11/30: PS, PDF.

### Lectures

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

• 8/29: Motivational lecture: algebraic geometry through the lens of elliptic curve theory (PS, PDF) Purely for motivational purposes, this will not be considered part of the course material. Introduces notions of smoothness, compactification, intersection theory, study of points over different fields, moduli problems, and other odds and ends.
• 8/31: Introduction to sheaves and schemes: classical affine algebraic sets and their relationship to ideals, presheaves, sheaves, morphisms and direct images (pushforwards) of sheaves, affine schemes, schemes as ringed spaces. (Portions of Hartshorne, sections II.1 and II.2, with some motivation from I.1. See also notes motivating the definition of a scheme: PS, PDF)
• 9/5: More on sheaves and schemes: stalks of sheaves, sheafification, schemes as locally ringed spaces, morphisms of schemes. (Further portions of Hartshorne, sections II.1 and II.2)
• 9/7: Discussion of schemes: examples of schemes and morphisms, discussion of morphisms and morphisms over a base, overdue proofs of propositions (omitted). (Further portions of Hartshorne, section II.2. See also notes on points and morphisms from the classical and scheme-theoretic perspectives: PS, PDF)
• 9/12: More on morphisms of affine varieties. Projective varieties.
• 9/14: Morphisms of projective varieties. Projective schemes. (Remainder of Hartshorne, section II.2)
• 9/19: An introduction to moduli spaces, universal properties, and representable functors. (See notes on Yoneda's lemma and representable functors: PS, PDF)
• 9/21: Existence of the fiber product, using Zariski sheaves and representable open subfunctors. (See notes on Zariski sheaves and representability of the fiber product: PS, PDF)
• 9/26: How to think about fiber products. Finiteness properties of schemes and morphisms. (Remainder of Hartshorne, section II.3)
• 9/28: Fixing the Zariski topology: properness and separatedness. (Portions of Hartshorne, section II.4. See also notes on understanding separatedness and properness: PS, PDF)
• 10/03: Immersions. Basic properties of separatedness and properness. Statement of valuative criteria. (More of Hartshorne, section II.4)
• 10/05: Proofs of valuative criteria. Proof that projective space is proper. (Remainder of Hartshorne, section II.4)
• 10/10: Introduction to sheaves of modules. Quasi-coherent and coherent sheaves. (Portions of Hartshorne, section II.5)
• 10/12: More on quasi-coherent sheaves and ideal sheaves. Invertible sheaves and maps to projective space. Grassmannians.
• 10/17: Quasi-coherent sheaves and Proj. More on maps to projective space. (Remainder of Hartshorne, section II.5)
• 10/19: Weil divisors, the divisor class group. (Portions of Hartshorne, section II.6)
• 10/24: More on DVRs and Weil divisors. (Portions of Hartshorne, section II.6)
• 10/26: Computations of Weil divisor class groups. (More of Hartshorne, section II.6)
• 10/31: Cartier divisors. (Yet more of Hartshorne, section II.6)
• 11/2: Cartier divisors and invertible sheaves. (Rest of Hartshorne, section II.6, except portion on curves)
• 11/7: Lines on cubic surfaces by degenerations (sketch). An introduction to dimension theory on schemes. (See notes on dimension theory for schemes: PS, PDF)
• 11/9: More on dimension theory. An introduction to flatness. (Portions of Hartshorne, section III.9)
• 11/14: More on flatness. An introduction to smooth, unramified, and etale morphisms. (More of Hartshorne, section III.9)
• 11/16: The sheaf of differentials.
• 11/21: Miscellaneous topics related to differentials and smoothness.
• 11/23: Thanksgiving: no class.
• 11/28: Berkeley-Stanford joint algebraic geometry colloquium at Stanford: no class.
• 11/30: Introduction to deformation theory and applications. (See notes on deformation theory: PS, PDF)
• 12/5: More on deformation theory and applications.
• 12/7: Deformations of smooth varieties and Cech cohomology.

### 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

• Properties of schemes (PS, PDF)
• Properties of morphisms (PS, PDF)