Course Announcement - Spring 2019

Math 255: Algebraic Curves

Instructor: Bernd Sturmfels

Time and Place: Tuesdays and Thursdays, 9:30-11, 3 Evans Hall

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 González-Diez: Introduction to Compact Riemann Surfaces and Dessins d’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