Explicit Models for Threefolds Fibred by K3 Surfaces of Degree Two

TL;DR

This paper studies threefolds fibred by degree two K3 surfaces, showing how their relative log canonical models can be explicitly reconstructed from specific data, and establishing a converse correspondence.

Contribution

It provides an explicit reconstruction method for the relative log canonical models of such threefolds and proves a converse result linking data sets to threefolds.

Findings

Explicit reconstruction of relative log canonical models

Existence of models from specified data sets

Bidirectional correspondence between data and threefolds

Abstract

We consider threefolds that admit a fibration by K3 surfaces over a nonsingular curve, equipped with a divisorial sheaf that defines a polarisation of degree two on the general fibre. Under certain assumptions on the threefold we show that its relative log canonical model exists and can be explicitly reconstructed from a small set of data determined by the original fibration. Finally we prove a converse to the above statement: under certain assumptions, any such set of data determines a threefold that arises as the relative log canonical model of a threefold admitting a fibration by K3 surfaces of degree two.