Stability of the Poincar\'e bundle

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