Geometric stability conditions under autoequivalences and applications: Elliptic surfaces

On a Weierstraß elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of Stab(X), and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects.