  We study the deformations of the minimally elliptic surface singularity
$N_{16}$. A standard argument reduces the study of the deformations of $N_{16}$
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 $N_{16}$. 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 $N_{16}$.
