Stability conditions on surfaces and contractions of curves

We study Bridgeland stability conditions on smooth surfaces arising from birational morphisms $S\to T$ to a singular surface. Assuming that $T$ has only ADE singularities or certain cyclic quotient singularities, we produce pre-stability conditions on $S$ whose central charges depend on the pullback of an ample divisor on $T$. Moreover, we prove the support property for the cyclic quotient case (including the $A_n$ singularities). These stability conditions arise as limits of the Arcara-Bertram stability conditions on $S$. In a complementary direction, we study birational maps $S\to T$ for which a stability condition $S$ can be obtained using the pullback of an ample class on $T$. We prove that the morphism cannot contract smooth curves of positive genus.