Degenerations of $K3$ surfaces of degree $4$

Abstract:

A generic $K3$ surface of degree $4$ may be embedded as a nonsingular quartic surface in ${{\mathbf {P}}_3}$. Let $f:X \to \operatorname {Spec} \;{\mathbf {C}}[[t]]$ be a family of quartic surfaces such that the generic fiber is regular. Let ${\Sigma _0}$, ${\Sigma _2^0}$, ${\Sigma _4}$ be respectively a nonsingular quadric in ${{\mathbf {P}}_3}$, a cone in ${{\mathbf {P}}_3}$ over a nonsingular conic and a rational, ruled surface in ${{\mathbf {P}}_9}$ which has a section with self intersection $- 4$. We show that there exists a flat, projective morphism $f’:X’ \to {\text {Spec}}\;{\mathbf {C}}[[t]]$ and a map $\rho :{\text {Spec}}\:{\mathbf {C}}[[t]] \to {\text {Spec}}\:{\mathbf {C}}[[t]]$ such that (i) the generic fiber of $f’$ and the generic fiber of the pull-back of $f$ via $\rho$ are isomorphic, (ii) the fiber ${X’_0}$ of $f’$ over the closed point of ${\text {Spec}}\;{\mathbf {C}}[[t]]$ has only insignificant limit singularities and (iii) ${X’_0}$ is either a quadric surface or a double cover of ${\Sigma _0}$, ${\Sigma _2^0}$ or ${\Sigma _4}$. The theorem is proved using the geometric invariant theory.