Explicit models for threefolds fibred by \(K3\) surfaces of degree two (Q2841825)

The goal of this paper is to construct explicitly projective threefolds \(X\) admitting a fibration \(X \rightarrow C\) such that the general fibre is a \(K3\) surface of degree two. While interesting fibered threefolds of low degree are typically obtained as a complete intersection in a bundle of weighted projective spaces, the approach of this paper is to generalise a technique used by \textit{F. Catanese} and \textit{R. Pignatelli} [Ann. Sci. Éc. Norm. Supér. (4) 39, No. 6, 1011--1049 (2006; Zbl 1125.14023)] to construct fibrations of genus two curves. The general fibre of the fibration \(X \rightarrow C\) being a double cover of \(\mathbb P^2\) ramified over a sextic curve, the starting point of the construction is a fibration \(\mathcal P \rightarrow C\) of Veronese embedded projective planes degenerating in some points to a cone. The relative log canonical model of \(X \rightarrow C\) (polarised by a divisorial sheaf of relative degree \(2\)) is a double cover \(X^c \rightarrow \mathcal P\). The main theorem of this paper explains how some data associated to the polarised fibration \(X \rightarrow C\) determines the relative log canonical model and, vice versa, how this kind of data allows to construct \(X\).