Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic surfaces

TL;DR

This paper introduces a new limit of Bridgeland stability conditions on Weierstrass elliptic surfaces and studies how Fourier-Mukai transforms relate slope stable sheaves to these stability conditions.

Contribution

It defines a novel stability condition called $Z^l$-stability and analyzes the behavior of Fourier-Mukai transforms on slope stable sheaves within this framework.

Findings

Fourier-Mukai transforms map slope stable sheaves to $Z^l$-stable objects.

$Z^l$-semistable objects can be transformed into slope semistable torsion-free sheaves.

The work bridges classical stability with derived category stability conditions.

Abstract

On a Weierstra{\ss} elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted $Z^l$-stability, by varying the polarisation in the definition of Bridgeland stability along a curve in the ample cone of $X$. We show that a slope stable torsion-free sheaf of positive (twisted) degree or a slope stable locally free sheaf is taken by a Fourier-Mukai transform on $D^b(X)$ to a $Z^l$-stable object, while a $Z^l$-semistable object of nonzero fiber degree can be modified so that its inverse Fourier-Mukai transform is a slope semistable torsion-free sheaf.