Text Books: The following
two books are required for this class:
William Fulton:
Algebraic Curves. An Introduction to Algebraic Geometry,
Reprint of 1969 original, Addison-Wesley, 1989.
Frances Kirwan:
Complex Algebraic Curves, London Mathematical Society Student Texts, 23, Cambridge University Press, 1992.
Syllabus:
In September and October, we shall cover the core
material on curves from the two text books:
local properties, plane curves, morphism and rational maps,
Riemann surfaces, differentials, Puiseux series,
resolution of singularities, and the Riemann-Roch Theorem.
In November, we shall discuss selected topics,
with emphasis on
19th century geometry,
algorithms, moduli spaces, and tropical curves.
Term Papers:
Students wrote term papers on topics
of their choice related to algebraic curves.
Here are links that show their excellent accomplishments:
Emily Berger:
Chip Firing and Riemann-Roch
Sarah Brodsky:
Inflection Points of Real and Tropical Curves
Ka Laam Chan:
Resolution of Singularities
Shelly Manber:
Algebraic Geometry Codes
James McIvor:
A First Glimpse of Deformation Theory
Maria Monks:
Duality of Plane Curves
Ralph Morrison: Tropical Intersections: Where they go wrong and where they go right
Qingchun Ren:
A Formula for the Cayley-Bacharach Theorem
Jose Rodriguez:
How to E-Mail a Riemann Surface
Zvi Rosen:
Graph Curves
Eugenia Rosu: Reduction of Ternary Forms
Shamil Shakirov:
Invariants of Quartic Curves
Alexander Shapiro:
Harnack Curves
Gus Schrader:
Faye's Trisecant Identity
Victoria Wood:
Parametrizing Rational Curves
Raki Yatchak:
The Degree of a Severi Variety
Consultant: Melody Chan will help with the course. Questions can be directed to either her 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:
August 25: Foundations [Kirwan, Chapter 2]
August 30: Bezout's Theorem [Kirwan, Section 3.1]
September 1: Bezout's Theorem [Kirwan, Section 3.1]
September 6: Points of inflection and
cubic curves [Kirwan, Section 3.2]
September 8: The degree-genus formula [Kirwan, Section 4.1]
September 13: No Class: please consider attending the POLYMAKE seminar
September 15: Branched covers of the line [Kirwan, Sections 4.2-4.3]
September 20: The
Weierstrass p-function [Kirwan, Section 5.1]
September 22: Riemann surfaces [Kirwan, Section 5.2]
September 27: Holomorphic differentials [Kirwan, Section 6.1]
September 29: Abel's Theorem [Kirwan, Section 6.2]
October 4: Melody Chan:
The Riemann-Roch Theorem [Kirwan, Section 6.3]
October 6: Melody Chan: The Riemann-Roch Theorem [Kirwan, Section 6.3]
October 11: Local rings, DVRs, Multiplicities [Fulton, Sections 2.4, 2.5, 3.1, 3.2]
October 13: Linear Systems, Multiple Points, Noether's Theorem [Fulton, Sections 5.2, 5.4, 5.5]
October 18: Curves in Projective Space [Fulton, Chapters 4 and 6]
October 20: Charley Crissman: Introduction to Moduli of Curves
October 25: Resolution of Singularities [Fulton, Chapter 7]
October 27: Shaowei Lin:
Computing Resolutions -- How and Why
November 1: Divisors and their Sections [Fulton 8.1-8.2]
November 3: Riemann's Theorem, Derivations, Differentials [Fulton 8.3-8.4]
November 8: Canonical Divisors and Riemann-Roch revisited [Fulton 8.5-8.6]
November 10: David Eisenbud: The Most Interesting Embeddings of a Curve
November 15: Shamil Shakirov: Invariants of Quartic Curves
Jose Rodriguez: How to E-mail a Riemann Surface
November 17: Qingchun Ren: A Formula for the Cayley-Bacharach Theorem
Emily Berger: The Riemann-Roch Theorem for Graphs
November 22: Maria Monks: Duality of Plane Curves
Zvi Rosen: Graph Curves
November 29: Rika Yatchak: Severi Varieties
Sarah Brodsky: Inflection Points of Real and Tropical Plane Curves
December 1:
Shelly Manber: Algebraic Geometry Codes
Alexander Shapiro: Harnack Curves.
December 5: [10am-noon, 939 Evans]
Gus Schrader: Faye's Trisecant Identity
Ralph Morrison: Tropical Intersections -- Where They Go Wrong, and
Where They Go Right
Eugenia Rosu: Stoll's Reduction Theory for Ternary Forms
Homework:
In the first eight weeks there will
be five assignments, posted here in pdf format:
Homework 1 is due Thursday, September 1.
Homework 2 is due Tuesday, September 13.
Homework 3 is due Tuesday, September 27.
Homework 4 is due Tuesday, October 11.
Homework 5 is due Thursday, November 3.
Term paper deadlines:
Thursday, October 27: Project proposal is due
Thursday, December 1: Final term paper is due
Grading:
The course grade will be based on both the homework
and the term paper.
