## 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 MathM Space Vector Bundles

C Vector Bundles