Math 137 -- Algebraic geometry -- Spring 2020

Mondays and Wednesdays 01:30 PM - 02:45 PM SC 310

This class is an introduction to algebraic geometry. Some topics we will cover include Hilbert's Nullstellensatz, affine and projective varieties, plane curves, Bézout's Theorem, morphisms of varieties, divisors and linear systems on curves, Riemann-Roch Theorem.

Instructor: Brooke Ullery (bullery@math, office SC 503, office hours Mondays 11:30-12:30, Tuesdays 4-5, and by appointment)

Graduate course assistant: Yujie Xu (yujiex@math, office hours/recitation Thursdays 7:30-8:30 PM in SC411)

Course assistants: CJ Dowd (cjdowd@college, office hours Mondays 8-10 PM at Math night in Lev DHall) and Raphael Tsiamis (rtsiamis@college, office hours Tuesdays 8-10 in Eliot DHall)

Text: We will very roughly be following Fulton's Algebraic Curves, which is available for free (legally) here. However, there are many excellent introductory algebraic geometry texts that are worth taking a look at. Here are some others that you might find useful:

Homework: Problem sets will be assigned weekly (usually due Wednesdays). I encourage you to work on the problems together, but you must turn in your own solutions and list the names of your collaborators. At the end of the semester, I'll drop your lowest homework score.

Final presentation: CANCELED.

Grading: Homework: Now 100%


Assignments

You should submit each problem set on canvas by 11:59 PM the day it's due. I (and the CAs) strongly prefer that you tex your solutions, but I will also accept very neatly handwritten and scanned solutions.

Problem set 1: pdf file, tex file, due February 5
Problem set 2: pdf file, tex file, due February 12
Problem set 3: pdf file, tex file, due February 19
Problem set 4: pdf file, tex file, due February 26
Problem set 5: pdf file, tex file, due March 4
Problem set 6: pdf file, tex file, due March 11
Problem set 7: pdf file, tex file, due Friday, March 27
Problem set 8: pdf file, tex file, due Friday, April 3
Problem set 9: pdf file, tex file, due Friday, April 10
Problem set 10: pdf file, tex file, due Friday, April 17
Problem set 11: pdf file, tex file, due Wednesday, April 29

Lecture notes

Disclaimer: I am happy to share the lecture notes I write for the class, and I do my best to make them easy to read and to post them soon after I finish lecturing on each section. However, their primary purpose is for me to use as lecture notes. In particular, I make no promises as to when each section will be available, since some topics span several classes and I may not have a section fully written until we finish covering that topic. You should use these notes to supplement the class rather than solely relying on them.

Section 1: What is algebraic geometry?
Section 2: Algebraic sets
Section 3: The ideal of a subset of affine space
Section 4: Irreducibility and the Hilbert Basis Theorem
Section 5: Hilbert's Nullstellensatz
Section 6: Algebra detour
Section 7: Affine varieties and coordinate rings
Section 8: Regular maps
Section 9: Rational functions and local rings
Section 10: Affine plane curves
Section 11: Discrete valuation rings and multiplicities
Section 12: Intersection numbers
Section 13: Projective space
Section 14: Projective algebraic sets
Section 15: Homogeneous coordinate rings and rational functions
Section 16: Affine and projective varieties
Section 17: Morphism of projective varieties
Section 18: Projective plane curves
Section 19: Linear systems of curves
Section 20: Bézout's Theorem
Section 21: Abstract varieties
Section 22: Rational maps and dimension
Section 23: Rational maps of curves
Section 24: Blowing up a point in the plane