**Instructor: Matthew
Simpson**

Office: Herman Brown 447

**Office Hours:** Monday

Email: *hargis@rice.edu** **
*Phone extension: x2868

Course Time & Location: Wednesdays,

Prerequisites: Some basic knowledge of matrices will be needed as well as some mathematical sophistication and experience with proofs. A course in Linear algebra should suit these requirements well; a course in abstract algebra even better. If you haven't had a course in linear algebra but are interested in the class, email me. To some degree, the course will be tailored towards the participants, so hopefully anyone interested should be able to attend.

Goal: To get a working knowledge of plane curves and singularities and to apply this knowledge in the direction of your choice to an end-of-term project, thus getting a feel for mathematical research.

Description:

This seminar will focus on the computation of problems in Algebraic Geometry without a heavy emphasis on the theory of the subject. It is geared towards all undergraduates who are interested in exploring mathematical research or just want to learn about higher level math through a somewhat informal problem-solving approach.

Most of the semester will be spent gaining a working knowledge on various aspects of plane curves. This class is meant to be interactive. In general, half or more of each class will be spent presenting and discussing homework problems which were handed out the week before. The second half of the class will be spent introducing a new or related topic. Homework will be given out at the end of class or posted on the web page by the end of the week. The last few weeks of class will be devoted to student presentations.

Some probable topics to be covered include (in no particular order):

- singular locus of a
plane curve using Jacobian criterion
- multiplicity of a plane
curve singularity
- description of tangent
cones
- intersection
multiplicities of two plane curves meeting at the origin
- classification of plane
curves of small degree
- deciding
when two plane curve singularities are analytically equivalent
- the group law on cubics

**Grading: **

Attendance/Participation (70%): There will be weekly homework, to be presented at the beginning of class. Participation is a must.

Project (30%): You will prepare a project and a twenty-five minute presentation. We will discuss projects/project ideas half way through the semester.

Anyone interested in receiving more than one credit hour may be allowed to submit a written project along with their presentation. If you're interested in doing this, inform me as soon in the semester as reasonably possible.

**Referencs: **

- Robert Bix: Conics
and Cubics. Springer-Verlag. (The main, though not
complete, reference for the course. The library should have a
copy on reserve.)
- Shafarevich: Basic
Algebraic Geometry. Springer-Verlag.
- William Fulton:
*Algebraic Curves*. Addison Wesley. - Robin Hartshorne:
*Algebraic Geometry*. Springer-Verlag. - E. Brieskorn and H.
Knörrer:
*Plane Algebraic Curves*. Birkhauser.