## Math 256B - Section 1 - Algebraic Geometry

Instructor: Brian Osserman

Lectures: TuTh 2:00-3:30pm, Room 72 Evans

Course Control Number: 54982

Office: 767 Evans, e-mail:

Office Hours: Tu 1:00-2:00, W 2:00-3:00 (this week only: instead of W, office hour Th 11:30-12:30)

Prerequisites: 256A

Required Text: Hartshorne, Algebraic Geometry

Recommended Reading: Eisenbud and Harris, the Geometry of Schemes

Syllabus: This is the second semester of an integrated, year-long course in algebraic geometry. The primary source text will be Hartshorne's Algebraic Geometry. We will have already covered most of Chapter II in the fall. We will begin with the basic properties of sheaf cohomology and the statement of the Riemann-Roch theorem, and classical applications to the study of curves. This will serve as motivation for the development of cohomology in Chapter III. Some important topics from projective geometry that were skipped in the fall will be worked in as appropriate. Finally, we will conclude with some discussion of special topics, most likely involving moduli spaces.

Grading: 75% homework, 25% final paper

Homework: Homework will be assigned roughly weekly

### Welcome to Math 256B: Algebraic Geometry

Homework assignments and supplementary notes, as well as lecture synopses, will be posted here as the course progresses. For general comments, please see last semester's course web page.

### Problem sets

Problem sets will be posted here, due on Tuesdays. 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 unless they trivialize the problem or use machinery which we have not introduced.

• Problem set #1, due 2/6: PS, PDF.
• Problem set #2, due 2/15: PS, PDF.
• Problem set #3, due 3/1: PS, PDF. Note: although the due date of this problem set has been extended, a new problem set will be posted on 2/22, also due on 3/1. Act accordingly.
• Problem set #4, due 3/1: PS, PDF.
• Problem set #5, due 3/8: PS, PDF.
• Problem set #6, due 3/15: PS, PDF.
• Problem set #7, due 4/5: PS, PDF.

### Final paper guidelines

The purpose of the final paper is twofold: to learn in detail about a topic not covered in class, and to practice mathematical writing. As such, it will be graded as much on clarity of writing as on any other factors.

Guidelines:

• The paper is due on the final day of classes, 5/8. This is a hard deadline, as I will be out of the country shortly thereafter.
• Please come to my office hours during the week of 4/10 to discuss your choice of topic.
• The paper should be roughly 8-10 pages in default tex formatting. Using latex with the amsart documentclass is recommended, but if you are already familiar with another flavor of tex, you may use that instead.
• There is no requirement (nor encouragement) of originality of content, but of course the paper should be entirely in your own words.
• The paper should be fully self-contained, and written like a research article. In particular, it should have a brief introduction explaining the main material to be discussed, and ideally some explanation of why it is important.
• Material from lecture may be used without citation. Material closely related to but not covered in lecture should be used with a specific statement of the result needed, and a citation. Many of you will also have to use results beyond material from lecture. These should always include a specific statement and citation, and you should consult with me to make sure you have a reasonable balance between citations and included arguments.
Possible topics: you are free to choose any topic in algebraic geometry which interests you. If you have something in mind, that's great. If not, the following areas would offer plenty of good paper topics:
• Intersection theory
• The minimal model program
• The Weil conjectures
• The algebraic fundamental group
• Etale cohomology
• Group schemes
• Quot and Hilbert schemes
• Moduli spaces and stacks
• Embedded deformations
• Rational curves on varieties
• The interpolation problem
• Curves in projective space
• Limit linear series

### Lectures

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

• 1/16: Introduction to sheaf cohomology: overview of basic properties, Riemann-Roch and first applications to curves. (For a bare-bones summary, see cheat sheet on cohomology of sheaves: PS, PDF)
• 1/18: Criteria for closed immersions. Linear series. Applications to curves of genus 0 and 1. (See notes on closed immersions: PS, PDF)
• 1/23: Elliptic curves and the classical group law. (Portions of Hartshorne, section IV.4)
• 1/25: Elliptic curves and the group law via Pic. (Portions of Hartshorne, section IV.4)
• 1/30: Gonality of curves and the canonical imbedding. (Portions of Hartshorne, section IV.5)
• 2/1: Derived functor cohomology. (Hartshorne, section III.1)
• 2/6: Sheaf cohomology: first properties. (Hartshorne, section III.2)
• 2/8: Cohomology and affine schemes. (Hartshorne, section III.3)
• 2/13: Cech cohomology. (Hartshorne, section III.4)
• 2/15: Cohomology of projective space, and interlude on ampleness and very ampleness. (Portions of Hartshorne, sections III.5, II.5 and II.7)
• 2/20: Ampleness, very ampleness, and vanishing conditions. (more from the above sections of Hartshorne)
• 2/22: Ext. (Hartshorne, section III.6)
• 2/27: Serre duality I. (Hartshorne, portion of section III.7)
• 3/1: Serre duality II. (Hartshorne, more of section III.7)
• 3/6: Applications of cohomology: Hilbert polynomials. (See background notes on associated points: PS, PDF, and notes on Hilbert polynomials: PS, PDF)
• 3/8: More on Hilbert polynomials and degrees.
• 3/13: Rational maps, sheaf Proj, and blowups. (Primarily portions of Hartshorne, section II.7)
• 3/15: Remarks on Grassmannians and universal constructions. Smoothness in characteristic 0. (Primarily last portion of Hartshorne, section III.10)
• 3/20: Derived pushforwards, towards the theorem on formal functions. (Portions of Hartshorne, sections III.8, III.9, and III.11)
• 3/22: The theorem on formal functions, and Zariski's "main" theorem. (Portions of Hartshorne, section III.11)
• 4/3: More corollaries of the theorem on formal functions, and initial discussion of cohomology and base change. (Portions of Hartshorne, sections III.11 and III.12)
• Disclaimer: The lectures from this point on will be a series of informal seminar-style lectures on topics which I find interesting, but don't necessarily know much about. As such, I make no warranty as to the correctness, completeness, or cogency of any of the notes below.

• 4/5: An overview of intersection theory. (See accompanying notes: PS, PDF)
• 4/10: An overview of birational geometry and minimal models. (See accompanying notes: PS, PDF)
• 4/12: The Weil conjectures and etale cohomology. (See accompanying notes: PS, PDF)
• 4/17: The moduli schemes Quot, Hilb, and Hom. (See accompanying notes: PS, PDF)
• 4/19: Deformations of subschemes and morphisms. (See accompanying notes: PS, PDF)
• 4/24: Mori's bend-and-break argument. (See accompanying notes: PS, PDF)
• 4/26: Moduli spaces and algebraic stacks.
• 5/1: Stable curves and the irreducibility of the moduli space of curves.
• 5/3: Limit linear series.
• 5/8: Fundamental groups and anabelian geometry.

### 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.

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