Degenerations of K3 Surfaces of Degree Two

TL;DR

This paper classifies the possible singular fibers in semistable degenerations of degree two K3 surfaces, describing their images as specific hypersurfaces or complete intersections in weighted projective spaces.

Contribution

It provides a complete classification of the singular fibers in degenerations of degree two K3 surfaces and explicitly describes the associated hypersurfaces and complete intersections.

Findings

Fibers map to sextic hypersurfaces in P(1,1,1,3)

Fibers map to degree (2,6) complete intersections in P(1,1,1,2,3)

Full classification of possible singular fibers

Abstract

We consider a semistable degeneration of K3 surfaces, equipped with an effective divisor that defines a polarisation of degree two on a general fibre. We show that the map to the relative log canonical model of the degeneration maps every fibre to either a sextic hypersurface in P(1,1,1,3) or a complete intersection of degree (2,6) in P(1,1,1,2,3). Furthermore, we find an explicit description of the hypersurfaces and complete intersections that can arise, thereby giving a full classification of the possible singular fibres.