|
Public viva-voce Notification Speaker: Jagadish Pine Date: 26 April 2024 Time: 11:45 AM Venue: Seminar Hall A universal moduli space of stable parabolic vector bundles over the marked stable curves. Jagadish Pine Chennai Mathematical Institute. 26-04-24 Abstract Let X be a smooth curve degenerating to an irreducible nodal curve $X_0$ over a DVR Spec(A). Seshadri constructed a proper and flat degeneration of the moduli space $U_X$ of semistable vector bundles on X to the moduli space of semistable torsion-free sheaves on $X_0$. Gieseker constructed a different degeneration of $U_X$ in the case of rank 2 and odd degree d, proving a conjecture of Newstead-Ramanan on the vanishing of Chern classes of the tangent bundle of $U_X$. Nagaraj-Seshadri later generalized Gieseker's construction to the case of coprime rank r and degree d. In this talk, we will discuss a generalization of Gieseker-type construction to the parabolic case. We will discuss the construction of a universal moduli space of stable parabolic vector bundles $U_{g,n}$ over the moduli space of marked stable curves $\overline{M}_{_{g, n}}$. Under a coprimality assumption, the moduli space $U_{g,n}$ will be projective. In terms of singularities, the space $U_{g,n}$ provides significant improvement over its torsion-free counterparts. The total space of $U_{g,n}$ will have finite quotient singularities. For any singular curve in $\overline{M}_{_{g, n}}$ with a trivial automorphism group, the fiber will have a singularity that is a product of analytic normal crossings.
|
