Jagadish Pine
TIFR, Mumbai
January 18, 2024
A universal moduli space of stable parabolic vector bundles over the marked stable curves: 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 $d$.
TIFR, Mumbai
January 18, 2024
A universal moduli space of stable parabolic vector bundles over the marked stable curves: 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. The objects that appear over the boundary of $\overline{M}_{_{g, n}}$ that is, those over singular curves, will remain vector bundles. We will also talk about the singularities of the total space and of the fibers over the marked stable curves, which are "nicer" than their torsion-free
counterparts.