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. In the first half of this article, we define the notion of limit tilt stability, which is closely related to Bayer's polynomial stability. We 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 stable object. We also show that the inverse… CONTINUE READING