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

TL;DR

This paper investigates the existence of Poincare sheaves on moduli spaces of torsionfree sheaves over irreducible curves, establishing conditions based on coprimality that determine their existence.

Contribution

It provides a criterion involving coprimality of parameters that determines when Poincare sheaves exist on these moduli spaces.

Findings

Poincare sheaves exist when 2n is coprime to d + n(1 - genus)

No Poincare sheaves exist on any nonempty open subset when 2n is not coprime to the invariant

Results clarify the geometric structure of moduli spaces of sheaves over curves.

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 $\chi\, =\, d+n(1-\text{genus}(Y))$, 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 $U'_Y$.