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:

- Harris; Algebraic Geometry
- Smith; An Invitation to Algebraic Geometry
- Reid; Undergraduate Algebraic Geometry
- Shafarevich; Basic Algebraic Geometry I

**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%

**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

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