Poincare sheaves on the moduli spaces of torsionfree sheaves over an irreducible curve

Abstract

Let $Y$ be a geometrically irreducible reduced projective curve defined over real numbers. Let $U_Y$ (respectively, $U'_Y$) be the moduli space of geometrically stable torsionfree sheaves (respectively, locally free sheaves) on $Y$ of rank $n$ and degree $d$. Define $χ\, =\, d+n(1-\text{genus}(Y))$, where $\text{genus}(Y)$ is the arithmetic genus. If $2n$ is coprime to $χ$, then there is a Poincare sheaf over $U_Y\times Y$. If $2n$ is not coprime to $χ$, then there is no Poincare sheaf over any nonempty open subset of $U'_Y$.