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

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