Line configurations and K3 surfaces

Configurations are combinatorial data that encode incidences of points and lines in the plane. I will give an argument that one can expect the moduli space of realizations of an n_3 configuration to be a Calabi-Yau variety. In particular, 10_3 configurations should correspond to K3 surfaces. The realization spaces of certain 10_3 configurations can be described concretely as elliptic surfaces with Du Val singularities. Resolving these singularities yields K3 surfaces, as predicted, and keeping track of singular fibers allows us to compute enough standard invariants to uniquely identify them. The same surfaces can be reconstructed using geometric invariant theory, endowing them with an elegant functorial interpretation.