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

**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: 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

**Homework:**
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