Commutative Algebra (Math 250B). Experience in working with Fields, Rings, Modules, Ideals, and their Gröbner Bases.

**Text Books:** The following
two text books will be used in this class:

Frances **Kirwan**:
Complex Algebraic Curves, London Mathematical Society Student Texts, 23, Cambridge University Press, 1992.

William **Fulton**:
Algebraic Curves. An Introduction to Algebraic Geometry,
Reprint of 1969 original, Addison-Wesley, 1989.

**Syllabus:**
Before Spring Break, we will cover the core
material on curves from the two text books:
local properties, plane curves,

morphisms and rational maps,
Riemann surfaces, differentials, Puiseux series,
resolution of singularities, and the Riemann-Roch Theorem.

After Spring Break, students and instructor will
present selected topics (e.g. 19th century geometry,
algorithms, moduli, and tropical curves).

**Term Papers:**
Students select a topic of their
choice related to algebraic curves. They will research that

topic and
write a term paper about their findings.
Presentations on these projects will take place in April.

**Grading:**
The course grade will be based on the homework (50%) and the term paper (50%).

**Consultants:**
Madeline Brandt
and Lynn Chua
will help with the course.
Questions can be directed to either them or
me.

**Further Reading:**
Here is a selection of recommended resources on algebraic curves:

Lecture Notes from the Math 255 class taught by **Hendrik Lenstra**
in the Fall of **1995**.

Egbert Brieskorn and Horst Knorrer: Plane Algebraic Curves,
Birkhauser Verlag, Basel, 1986.

Joe Harris and Ian Morrison: Moduli of Curves,
Graduate Texts in Mathematics, 187, Springer 1998.

George Salmon, Arthur Cayley:
A Treatise on the Higher Plane Curves,
Elibron Classics, original from 1852.

Rafael Sendra, Franz Winkler and Sonia
Perez-Diaz: Rational Algebraic Curves - A Computer Algebra
Approach, Springer, 2008.

Ernesto Girondo and Gabino GonzĂˇlez-Diez:
Introduction to Compact Riemann Surfaces and Dessins dâ€™Enfants,
Cambridge University Press, 2011.

** Schedule:**

January 22: Foundations [Kirwan, Chapter 2]

January 24: Bezout's Theorem [Kirwan, Section 3.1]

January 29: Points of inflection and cubic curves [Kirwan, Section 3.2]

January 31: The degree-genus formula [Kirwan, Section 4.1]

February 5: Branched covers of the line [Kirwan, Sections 4.2-4.3]

February 7:
Invariant Theory of Plane Curves

February 12: The
Weierstrass p-function [Kirwan, Section 5.1]

February 14: Riemann surfaces [Kirwan, Section 5.2]

February 19 [MB]: Holomorphic differentials [Kirwan, Section 6.1]

February 21 [MB]: Abel's Theorem [Kirwan, Section 6.2]

February 26: The Riemann-Roch Theorem [Kirwan, Section 6.3]

February 28: The Riemann-Roch Theorem [Kirwan, Section 6.3]

March 5: Local rings, DVRs, Multiplicities [Fulton, Sections 2.4, 2.5, 3.1, 3.2]

March 7: Linear Systems, Multiple Points, Noether's Theorem [Fulton, Sections 5.2, 5.4, 5.5]

March 12: Varieties, Morphisms, and Rational Maps [Fulton, Chapter 6]

March 14: Resolution of Singularities [Fulton, Chapter 7]

March 19 [LC]: Divisors, Riemann's Theorem, Differentials [Fulton 8.1-8.4]

March 21 [LC]: Canonical Divisors and Riemann-Roch [Fulton 8.5-8.6]

**Student Lectures:**

April 2: The Plücker Formulas [Yuhan Jiang]

April 2: Inflection Points of Plane Cubics [Tyler Zhu]

April 4: Orthogonal Matrices with Maximal 4-Norm [Zitong Yang]

April 4: Riemann-Roch on Graphs [Frederick Huang]

April 9: Riemann-Roch in the 20th Century [Siyang Liu, Zhongkai Tao]

April 9: Riemann-Roch for Algebraic Surfaces [Nikolay Grantcharov, Sanat Mulay]

April 11: Algebraic Geometry Codes [Siqi Liu, Peter Manohar, Tahsin Saffat]

April 11: Elliptic Curve Cryptography [Andrew Gitlin, Kristina Nelson, Jana Sotakova]

April 16: Completions of Rings and Curve Singularities [Marvin Castellon]

April 16: Chow Rings [Holly Mandel]

April 18: Differential Forms on Riemann Surfaces [Suxuan Chen]

April 18: Belyi's Theorem [Emilio Valle]

April 23: Modular Curves and Modularity Theorems [Zhenghui Li]

April 23: Mordell-Weil Theorems [Grant Posner]

April 23: Nonnegative Polynomials and Sums of Squares [Han Feng, 11:10h in 939 Evans]

April 25: Moduli Spaces of Riemann Surfaces [Ziwen Zhao]

April 25: Moduli Spaces of Stable Maps [Foster Tom]

**Workshops:**

April 30-May 3:
Hyperbolic Polynomials at the Simons Institute

May 6-May 10:
Moduli Spaces at MSRI

**Homework:**
There are seven assignments. Click on the date to see solutions:

**due ****January 29:** Kirwan 2.2, 2.4, 2.5, 2.7, 2.8, 3.1, 3.6

**due February 5:** Kirwan 3.3, 3.8, 3.11, 3.13, 3.14, 3.16

**due February 12:** Kirwan 4.1, 4.2, 4.3, 4.4, 4.5

**due February 19:** Kirwan 5.4, 5.9, 5.10, 5.12, 5.14, 5.18

**due February 26:** Kirwan 6.1, 6.3, 6.5, 6.6, 6.7, 6.8

(Problem 6.3 has a typo: one occurrence of "meromorphic" should be
"holomorphic")

**due March 5:** Kirwan 6.10, 6.11, 6.15 and Fulton 2.17, 8.2, 8.6

(Problem 6.15: must assume that the curve has genus one)

**due March 12:** Fulton 2.25, 2.28, 3.6, 3.14, 5.11, 5.19, 5.21, 5.21, 5.30

** Term paper deadlines:**

Thursday, March 14: Project proposal is due

Tuesday, May 14: Final term paper is due