Poincaré Sheaves on the Moduli Spaces of Torsionfree Sheaves over an Irreducible Curve

Abstract

Let Y be a geometrically irreducible reduced projective curve defined over ℝ. Let U Y (n, d) (respectively, ) 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 − genus(Y)), where genus(Y) is the arithmetic genus. If 2n is coprime to χ, then there is a Poincaré sheaf over U Y (n, d) × Y. If 2n is not coprime to χ, then there is no Poincaré sheaf over any nonempty open subset of .