Deformations of singularities and variation of git quotients

We study the deformations of the minimally elliptic surface singularity N16. A standard argument reduces the study of the deformations of N16 to the study of the moduli space of pairs (C,L) consisting of a plane quintic curve and a line. We construct this moduli space in two ways: via the periods of K3 surfaces and by using geometric invariant theory (GIT). The GIT construction depends on the choice of the linearization. In particular, for one choice of linearization we recover the space constructed via K3 surfaces and for another we obtain the full deformation space of N16. The two spaces are related by a series of explicit flips. In conclusion, by using the flexibility given by GIT and the standard tools of Hodge theory, we obtain a good understanding of the deformations of N₁₆.