Stability of the Poincaré bundle

Abstract

Let $C$ be a nonsingular projective curve of genus $g\ge2$ defined over the complex numbers, and let $M_ξ$ denote the moduli space of stable bundles of rank $n$ and determinant $ξ$ on $C$, where $ξ$ is a line bundle of degree $d$ on $C$ and $n$ and $d$ are coprime. It is shown that the universal bundle $\cu_ξ$ on $C\times M_ξ$ is stable with respect to any polarisation on $C\times M_ξ$. It is shown further that the connected component of the moduli space of $\cu_ξ$ containing $\cu_ξ$ is isomorphic to the Jacobian of $C$.