Math 499: VIGRE Computational Algebraic Geometry Junior Seminar (Spring 2006)  

       

                                A pair of elliptic curves.


Instructor: Matthew Simpson
Office: Herman Brown 447
Office Hours: Monday 3:00-4:00 pm, Tuesday 1:00-2:30pm, Wednesday 2:30-3:30, and by appointment
Email: hargis@rice.edu
Phone extension: x2868
Course Time & Location: Wednesdays, 12:00-12:50, Herman Brown 453  

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.