Boston and Tufts Universities Seminar on Vector Bundles on Curves

Fall Semester Organized by Emma Previato, Montserrat Teixidor and Loring Tu

Spring semester Organized by Montserrat Teixidor

Meets at PSY53 Psychology Building, Basement 53, Boston University, 64-72-86 Cummington St. [BU Campus Map]
Room 1, Mathematics Department Department, Bromfield Pearson Buildijg [Contact List]

Bibliography

  • P.Newstead Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental reseach, Lecture notes in Mathematics and Physics 51, 1978 [link]
  • S.Mukai An introduction to invariants and moduli. Translated from the 1998 and 2000 Japanese editions by W. M. Oxbury. Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003. [link]

Some elementary notes

[PDF] Vector Bundles on Curves

Some Abstracts


9/21 This lecture will cover basic properties of vector bundles including stability. I will present vector bundles as extensions of smaller rank bundles and show applications.

10/12 We will discuss the concept of quotient (orbit space, categorical, geometric and good quotients) and its use in the construction of moduli spaces. We will describe in particular the construction of affine and projective quotients using geometric invariant theory. The content will be similar to that of Lecture 1 of my "Polish" notes (see http://www.mimuw.edu.pl/~jarekw/EAGER/Lukecin02).

2/23 Title: Dolgachev's conjecture on the moduli space of rank 3 bundles on a genus 2 curve. Abstract: For a curve of genus 2, the moduli space of rank 3 vector bundles with trivial determinant can be realized as a double cover of the 8-dimensional projective space branched along a sextic hypersurface. On the other hand, Coble proved a century ago that there exists a unique cubic hypersurface in 8-space which is (1) singular along the Jacobian variety of the curve (which is embedded using the system of 3-theta functions) and (2) invariant under the natural action coming from torsion points of order 3 on the Jacobian. Dolgachev has made the striking conjecture than these two seemingly completely unrelated hypersurfaces are in fact projectively dual to each other. I will discuss the background of the problem and present a proof of Dolgachev's conjecture.

Date Time Location Speaker Title
Tuesday Sept. 5 2:00pm BU Emma Previato (BU) Introduction
Tuesday Sept. 12 2:00pm BU Emma Previato (BU) Jacobians
Thursday Sept. 21 2.30 Tufts Montserrat Teixidor (Tufts) Basic Properties of Vector Bundles and Extensions
Thursday Sept. 28 2.30 Tufts Montserrat Teixidor (Tufts) Vector Bundles on Rational and Elliptic Curves
Thursday Oct 5 NO SEMINAR CLAY Workshop starts tomorrow
Thursday Oct 12 2.30pm BU Peter Newstead (Liverpool, UK) Quotients and Moduli (GIT)
Thursday Oct 19 2.30pm Tufts Peter Newstead (Liverpool, UK) Construction of moduli of Vector Bundles on Curves
Thursday Oct 26 2.30pm BU Peter Newstead (Liverpool, UK) Geometry of moduli spaces
Thursday November 2 2:30pm Tufts Montserrat Teixidor (Tufts) Vector Bundles on Singular curves
Thursday November 9 2:30pm Tufts Montserrat Teixidor (Tufts) Brill-Noether and limit linear series for vector bundles
Thursday November 30 2:30pm BU Steven Rosenberg (BU) Chern classes and Riemann-Roch Theorem
Thursday December 7 2:30pm Tufts Montserrat Teixidor (Tufts) Theta divisors for Vector Bundles
Thursday January 25 3:30pm Tufts Sebastian Casalaina-Martin (Harvard) Some results of Laszlo on Tangent Cones to the Theta Divisor
Thursday February 1 3:30pm Tufts Sebastian Casalaina-Martin (Harvard) Some results of Laszlo on Tangent Cones to the Theta Divisor II
Thursday February 8 3:30pm Tufts Sebastian Casalaina-Martin (Harvard) Some results of Laszlo on Tangent Cones to the Theta Divisor III
Wednesday February 14 4pm Tufts Montserrat Teixidor (Tufts) Base Locus for the Theta Linear System for Vector Bundles
Friday February 23 5.30pm Tufts Angela Ortega (Morelia, Mexico) Dolgachev Conjecture on the moduli space of rank three bundles on a curve of genus two
Wednesday March 7 4pm Tufts Cristian Gonzalez-Martinez (Tufts) Spectral curves
Wednesday March 14 4pm Tufts Cristian Gonzalez-Martinez (Tufts) Spectral curves and Vector Bundles
Wednesday April 4 4pm Tufts Cristian Gonzalez-Martinez (Tufts) Spectral curves and the Theta divisor
Friday April 13 4pm Tufts Cristian Gonzalez-Martinez (Tufts) Spectral curves and the Theta divisorII
Wednesday April 18 4pm Tufts Tawanda Gwena (Tufts) Maps between moduli spaces of curves
Wednesday April 25 4pm Tufts Tawanda Gwena (Tufts) Examples of base locus of the theta divisor

Other useful websites

Clay Math
M Space Vector Bundles
C Vector Bundles