Degenerations of 𝐾3 surfaces of degree 4

Abstract

A generic K 3 K3 surface of degree 4 4 may be embedded as a nonsingular quartic surface in P 3 {{\mathbf {P}}_3} . Let f : X → Spec C [ [ t ] ] f:X \to \operatorname {Spec} \;{\mathbf {C}}[[t]] be a family of quartic surfaces such that the generic fiber is regular. Let Σ 0 {\Sigma _0} , Σ 2 0 {\Sigma _2^0} , Σ 4 {\Sigma _4} be respectively a nonsingular quadric in P 3 {{\mathbf {P}}_3} , a cone in P 3 {{\mathbf {P}}_3} over a nonsingular conic and a rational, ruled surface in P 9 {{\mathbf {P}}_9} which has a section with self intersection − 4 - 4 . We show that there exists a flat, projective morphism f ′ : X ′ → Spec C [ [ t ] ] f’:X’ \to {\text {Spec}}\;{\mathbf {C}}[[t]] and a map ρ : Spec C [ [ t ] ] → Spec C [ [ t ] ] \rho :{\text {Spec}}\:{\mathbf {C}}[[t]] \to {\text {Spec}}\:{\mathbf {C}}[[t]] such that (i) the generic fiber of f ′ f’ and the generic fiber of the pull-back of f f via ρ \rho are isomorphic, (ii) the fiber X 0 ′ {X’_0} of f ′ f’ over the closed point of Spec C [ [ t ] ] {\text {Spec}}\;{\mathbf {C}}[[t]] has only insignificant limit singularities and (iii) X 0 ′ {X’_0} is either a quadric surface or a double cover of Σ 0 {\Sigma _0} , Σ 2 0 {\Sigma _2^0} or Σ 4 {\Sigma _4} . The theorem is proved using the geometric invariant theory.