Stability of the Parabolic Picard Sheaf

TL;DR

This paper proves the stability of the parabolic Picard sheaf associated with moduli spaces of stable parabolic vector bundles on algebraic curves, under certain coprimality conditions.

Contribution

It establishes the stability of the parabolic Picard sheaf on moduli spaces of stable parabolic bundles, a result not previously known.

Findings

Parabolic Picard sheaf is stable under specified conditions.

The stability result applies to moduli spaces with coprime invariants.

Provides foundational insight into the geometry of parabolic bundle moduli.

Abstract

Let $X$ be a smooth irreducible complex projective curve of genus $g\,\geq\, 2$, and let $D\,=\,x_1+\dots+x_r$ be a reduced effective divisor on $X$. Denote by $U_{\alpha}(L)$ the moduli space of stable parabolic vector bundles on $X$ of rank $n$, determinant $L$ of degree $d$ with flag type $\{\{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r$. Assume that the greatest common divisor of the collection of integers $\{\text{degree}(L),\, \{\{k^i_j\}_{j=1}^{m_i}\}_{i=1}^r\}$ is $1$; this condition ensures that there is a Poincar\'e parabolic vector bundle on $X\times U_{\alpha}(L)$. The direct image, to $U_{\alpha}(L)$, of the vector bundle underlying the Poincar\'e parabolic vector bundle is called the parabolic Picard sheaf. We prove that the parabolic Picard sheaf is stable.