Fourier-Mukai Transforms and Stable Sheaves on Weierstrass Elliptic Surfaces

On a Weierstraß 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.