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.
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).
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 both the homework (40%) and the term paper (60%).
Consultants: Madeline Brandt and Lynn Chua will help with the course. Questions can be directed to either them or me.
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.
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: The Weierstrass p-function [Kirwan, Section 5.1]
February 12: Riemann surfaces [Kirwan, Section 5.2]
February 14: Holomorphic differentials [Kirwan, Section 6.1]
February 19: Abel's Theorem [Kirwan, Section 6.2]
February 21: The Riemann-Roch Theorem [Kirwan, Section 6.3]
February 26: The Riemann-Roch Theorem [Kirwan, Section 6.3]
February 28: Local rings, DVRs, Multiplicities [Fulton, Sections 2.4, 2.5, 3.1, 3.2]
March 5: Linear Systems, Multiple Points, Noether's Theorem [Fulton, Sections 5.2, 5.4, 5.5]
March 7: Curves in Projective Space [Fulton, Chapters 4 and 6]
March 12: Resolution of Singularities [Fulton, Chapter 7]
March 14: Divisors and their Sections [Fulton 8.1-8.2]
March 19: Riemann's Theorem, Derivations, Differentials [Fulton 8.3-8.4]
March 21: Canonical Divisors and Riemann-Roch revisited [Fulton 8.5-8.6]
April 2: TBA
April 4: TBA
April 9: Student presentations
April 11: Student presentations
April 16: Student presentations
April 18: Student presentations
April 23: Student presentations
April 25: Student presentations
In the first eight weeks, there will
be five assignments:
due January 29: Kirwan 2.2, 2.4, 2.5, 2.7, 2.8, 3.1, 3.6
Term paper deadlines:
Thursday, March 14: Project proposal is due
Tuesday, May 14: Final term paper is due