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.Comment: 32 pages, comments welcome