Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces

Abstract

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.

arXiv comment45 pages. Exposition improved. This article significantly expands and supersedes arXiv:1710.04652. The notation and preliminaries also draw heavily from arXiv:1710.03771

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• A note on Bridgeland stability conditions and Catalan numbers
• Weight functions, tilts, and stability conditions
• Contractibility of space of stability conditions on the projective plane via global dimension function
• A framework for torsion theory computations on elliptic threefolds