Course: MA746, Algebraic Geometry II
Instructor: Jared Weinstein
Lectures: MWF 2:00 pm - 3:00 pm in MCS B31
Office Hours: M 10:00 am - 12:00 pm and Th 2:00 pm - 3:00 pm in MCS 227

Course Overview

This course is part two of an introduction to the theory of algebraic geometry. We will develop the modern language of schemes due to Grothendieck. This theory is technically demanding and requires much in the way of commutative algebra, but the payoff is very large. For instance, schemes enable us to work over non-algebraically closed fields, and also to deal with entities which are not reduced (such as the intersection of a plane conic with one of its tangent lines).

The plan is to cover most of Chapter II of Hartshorne's Algebraic Geometry book, plus some of Chapter IV. Topics will include sheaves, schemes, affine and projective spaces, morphisms, coherent sheaves, algebraic curves, and the Riemann-Roch theorem.

Prerequisites

Some commutative algebra: Rings, ideals, and modules. Having taken the first half of the Algebraic Geometry sequence is going to be very helpful, but probably not essential.

Assignments and Exams

Problem sets will be assigned at the rate of somewhat less than one per week. Two of the assignments will be elevated to the status of exam. The grading scheme is as follows: Ordinary HWs 50%, Midterm 25%, Final 25%. Assignments will appear on this page. These are to be turned in to me during lecture, or in my mailbox, or e-mailed to me as a PDF. On ordinary HWs, Working together is encouraged, but everything you turn in must be in your own words. Exams must be completed on your own.

HW #1, due Fri. 1/27: Ch II, 1.3, 1.19b, 1.22

HW #2, due Fri. 2/3: Ch II, 2.1, 2.l0, 2.19

HW #3, due Fri. 2/10: Ch II, 2.3, 3.1, 3.4, 3.6

HW #4, due Wed. 2/22: Ch II, 3.9, 3.10, 4.1, 4.2

HW #5, due Fri. 3/2: Ch II, 5.2, 5.5, 5.7, 5.12

Midterm, due Wed. 3/21.

HW #6, due Fri. 4/6: On 3/28 we will show that a nonsingular projective plane curve of degree d has genus (d-1)(d-2)/2. (a) Let X be a nonsingular curve which is a closed subvariety of P^1 x P^1. Let (d,e) be the class of X as a divisor in Pic(P^1 x P^1) = Z x Z. Show that the genus of X is (d-1)(e-1). (b) (Harder) Let X be a nonsingular curve in projective 3-space which is formed by the intersection of two surfaces of degrees d and e respectively. Then the genus of X is de(d+e-4)/2 + 1.

HW #7, due Fri. 4/27: Ch III, 4.3, 4.5, 4.7

Final, due Tues. 5/8.

Text

Robin Hartshorne, Algebraic Geometry, Springer GTM. Some files relevent to this course are here.