Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces

TL;DR

This paper introduces a new limit of Bridgeland stability conditions on Weierstrass elliptic surfaces, explores Fourier-Mukai transforms' effects on stability, and studies wall-crossing phenomena for sheaves.

Contribution

It defines the $Z^l$-stability condition, characterizes Fourier-Mukai transforms' impact on stability, and analyzes wall-crossing for sheaves on elliptic surfaces.

Findings

Fourier-Mukai transforms relate slope stability to $Z^l$-stability.

Conditions identified for transforms to preserve stability.

Wall-crossing behavior for sheaves is characterized.

Abstract

On a Weierstra{\ss} elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show that 1-dimensional twisted Gieseker semistable sheaves are taken by a Fourier-Mukai transform to Bridgeland semistable objects.