Projective Poincaré and Picard bundles for moduli spaces of vector bundles over nodal curves

Abstract

Let $U_L^{{}^{\prime }s}(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y\times U_L^{{}^{\prime }s}(n,d)$ and the projective Picard bundle on $U_L^{{}^{\prime }s}(n,d)$ are stable for suitable polarisation. For a nonsingular point $x\in Y$, we show that the restriction of the projective Poincare bundle to $x\times U_L^{{}^{\prime }s}(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2,d$ odd, the Picard group of the moduli space $U_L^{\prime }(n,d)$ of semistable vector bundles of rank $n$ with determinant $L$ of degree $d$ is isomorphic to $\mathbb{Z}$.

29 pages, comments are welcome!