**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 GonzĂˇlez-Diez:
Introduction to Compact Riemann Surfaces and Dessins dâ€™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.