The zero locus of the polynomial y^2-x^3+3xt^2-2t^3 when t=0,2,4.

**Instructor:**Amanda Knecht

Office: Herman Brown 447

**Office Hours:** Monday 1:00-3:00 pm, Tuesday 11:00 am -noon, and by appointment

Email: knecht@rice.edu

Phone: 2868

Course Time & Location: Wednesdays, 12:00-12:50, Herman Brown 453

Prerequisites: Linear Algebra is the only suggested prerequisite of the course, but this is only suggested. If you have not had a course in Linear Algebra, email me and we will decide if you are ready for the seminar. knecht@rice.edu

Goal: This semester I hope to have everyone comfortable enough with plane curve singularities to be able to do computational research in the area. The projects at the end of the semester are meant to show you how much math you can learn in a few weeks, and how that math can be used to answer questions. Those wishing to participate in the summer VIGRE REU should be ready to dive into their research by the end of the semester.

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 one hour a week seminar.

The first weeks of the course will be devoted to establishing a confidence in the topics listed below. Students will present their work on the previous week's problems for the first 35-40 minutes of class. The last 10-15 minutes will be devoted to learning new material and assigning problems to be worked on for the next week. The last few weeks will be devoted to the students' presentations of work they have done outside of class.

Some topics that we will cover are:

- 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
- computing the tangent space to the versal deformation spaces of a plane curve singularity and a plane curve singularity with distinguished section
- computing the Milnor number of a plane curve singularity
- deciding when two plane curve singularities are analytically equivalent
- enumerating branches of a plane curve singularity
- writing out Puiseux series and Newton polygons of unibranch plane curves

**Assessment: **

Attendance/Participation (70%): This course is a seminar, so attendance is crucial. You will present problems at the beginning of each class.

Project (30%): You will prepare a project and a twenty-five minute presentation.

Anyone interested in receiving more than one credit hour can arrange to submit a written project along with their presentation.

**References:**

- William Fulton:
*Algebraic Curves*. Addison Wesley. - Robin Hartshorne:
*Algebraic Geometry*. Springer-Verlag. - E. Brieskorn and H. Knörrer:
*Plane Algebraic Curves*. Birkhauser.