Poincaré bundle for the fixed determinant moduli space on a nodal curve

Abstract

Let $Y$ be an integral nodal projective curve of arithmetic genus $g\ge 2$ with $m$ nodes defined over an algebraically closed field $k$ and $x$ a nonsingular closed point of $Y$. Let $n$ and $d$ be coprime integers with $n\ge 2$. Fix a line bundle $L$ of degree $d$ on $Y$. Let $U_Y(n,d,L)$ denote the (compactified) "fixed determinant moduli space". We prove that the restriction ${\mathcal{U}}_{L,x}$ of the Poincare bundle to $x\times U_Y(n,d,L)$ is stable with respect to the polarisation ${\theta }_L$ and its restriction to $x\times U_Y^{\prime }(n,d,L)$, where $U_Y^{\prime }(n,d,L)$ is the moduli space of vector bundles of rank $n$ and determinant $L$, is stable with respect to any polarisation. We show that the Poincaré bundle ${\mathcal{U}}_L$ on $Y\times U_Y(n,d,L)$ is stable with respect to the polarisation $a\alpha +b{\theta }_L$ where $\alpha$ is a fixed ample Cartier divisor on $Y$ and $a,b$ are positive integers.

References in corpus (1)

• Stability of the Poincaré bundle