We use orbifold structures to deduce degeneracy statements for holomorphic maps into logarithmic surfaces. We improve former results in the smooth case and generalize them to singular pairs. In particular, we give applications on nodal surfaces and complements of singular plane curves.
We prove that the bounded derived category of the surface constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category....