On the theta divisor of {${\rm SU}(r,1)$}

Abstract

Let $SU(r,1)$ be the moduli space of stable vector bundles, on a smooth curve $C$ of genus $g \geq 2$, with rank $r \geq 3$ and determinant $O_C(p)$, $p \in C$; let ${\cal L}$ be the generalized theta divisor on $SU(r,1)$. In this paper we prove that the map ${\phi}_{\cal L}$, defined by ${\cal L}$, is a morphism and has degree $1$.