Reider-type theorems on normal surfaces via Bridgeland stability

Using Langer's construction of Bridgeland stability conditions on normal surfaces, we prove Reider-type theorems generalizing the work done by Arcara-Bertram in the smooth case. Our results still hold in positive characteristic or when $\omega_X \otimes L$ is not necessarily a line bundle. They also hold when the dualizing sheaf is replaced by a variant arising from the theory of Du Bois complexes. For complex surfaces with at most rational double point singularities, we recover the optimal bounds for global generation and very ampleness as predicted by Fujita's conjecture.