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

Abstract

We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.

arXiv comment33 pages. To appear in J. Algebra. The previous version has been split into two parts, the first of which forms the basis of this version (dealing with transforms of torsion-free sheaves)