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Stability of the Parabolic Picard Sheaf (2405.18389v2)

Published 28 May 2024 in math.AG

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 ${{ki_j}{j=1}{m_i}}{i=1}r$. Assume that the greatest common divisor of the collection of integers ${\text{degree}(L),\, {{ki_j}{j=1}{m_i}}{i=1}r}$ 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.

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