Stability conditions on surfaces and contractions of curves

TL;DR

This paper constructs Bridgeland stability conditions on smooth surfaces related to birational morphisms to singular surfaces with ADE or cyclic quotient singularities, proving support properties and analyzing contraction restrictions.

Contribution

It introduces new stability conditions on surfaces from birational maps to singular surfaces, including support property proofs and contraction limitations for positive genus curves.

Findings

Constructed stability conditions depending on pullback of ample divisors.

Proved support property for cyclic quotient singularities.

Showed that morphisms cannot contract smooth curves of positive genus.

Abstract

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.