To celebrate New Year 2003 I propose the following project: to answer the basic questions in Brill-Noether theory for vector bundles on algebraic curves within 10 years, that is by

1 January 2013.

By basic questions, I mean: for a general curve C and for all triples of integers (n,k,d) with n greater than or equal to 2, k greater than or equal to1, to obtain the following information concerning the Brill-Noether locus B(n,d,k), consisting of all stable bundles of rank n, degree d, with at least k independent global sections:

  • (1) Is B(n,d,k) non-empty?
  • (2) Is B(n,d,k) connected, and, if not, what are its connected components?
  • (3) Is B(n,d,k) irreducible, and, if not, what are its irreducible components?
  • (4) What is the dimension (of each component) of B(n,d,k)?
  • (5) What is the singular set of B(n,d,k)?

  • For many (n,d,k), complete or partial information is known and I will try to post the latest information on this page.

    The project is an open one; all contributions are welcome (and will be needed) if we are to answer these questions. There are of course many further questions, for example to discuss the detailed geometry of the Brill-Noether loci, their classes in the Chow ring and/or cohomology ring of the moduli space, and the links with the moduli spaces of coherent systems, but I have chosen to concentrate on these basic ones to focus attention.

    The intention is that this page should be linked to other pages containing information on these problems. In the first instance, I would like to compile files of people who are interested in Brill-Noether theory and of papers, both with links where these are available. Please let me know by email if you would like to be included in the file of people and give me your web address so I can put in a link to your own webpage. (If you don't have a webpage or prefer not to be linked, I can include your email address.)

    For the time being, pending reconstruction of this page, see Vincent Mercat's presentation.

    Peter Newstead