Algebraic Geometry Seminar
Monday 30th Jan, 11:30 am
Venue: Ramanujan hall
Host: Sudarshan Gurjar
Speaker: Arusha
Affiliation: TIFR, Mumbai
Title: Poincare and Picard bundles for moduli spaces of vector bundles over nodal curves
Abstract: Poincare and Picard bundles and their different variants have been a topic of interest ever since the quest for moduli spaces of vector bundles was initiated, owing to their universality. Though a great deal is known about these objects in the case of smooth curves, the study on singular curves has been relatively slow. Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and difficult. It was known since late 1960s that there does not exist a Poincar´e bundle (a universal family) for the moduli problem of vector bundles on smooth curves if the rank and degree are not coprime. The primary aim of the talk is to discuss the non-existence of a Poincare bundle in the non-coprime case for nodal curves. There has also been ample interest to understand the stability of Poincar´e and projective Poincare bundles as well as Picard and projective Picard bundles. The secondary aim is to discuss the stability of projective Poincar´e and Picard bundles, again when the degree and rank are not relatively prime to each other in the context of nodal curves. On the way to achieve these goals, we compute the codimension of a few closed subsets of the moduli spaces. They are of independent interest and have other applications; we discuss a few of them. This is a joint work with Prof. Usha Bhosle and Dr. Sanjay Singh.