Degenerations of \(K3\) surfaces of degree two (Q2862129)

In the paper under review, the author studies a semistable degeneration \(\pi: X \rightarrow \Delta\) of \(K3\) surfaces with general fibre \(X_t\) equipped with a polarization of degree two.NEWLINENEWLINETo be more precise, the author works in the following setting. First, he assumes that \(\omega_X \cong \mathcal{O}_X\). Second, he supposes there exists an effective divisor \(H\) on \(X\), nef and flat over \(\Delta\) such that \(H\) induces a nef and big divisor \(H_t\) on \(X_t\), satisfying \(H^2_t=2\). Moreover, let \(\phi: X \rightarrow X^c\) be the morphism taking \(X\) to the relative log canonical model \(X^c\) of the pair \((X,H)\).NEWLINENEWLINEThe main result of the paper is a description of \(\phi(X_0)\). Indeed, there are only the following cases for \(\phi(X_0)\). {\parindent=6mm \begin{itemize} \item(Hyperelliptic case) A sextic hypersurface NEWLINE\[NEWLINE \{z^2-f_6(x_i)=0\} \subset \mathbb{P}_{(1,1,1,3)}[x_1,x_2,x_3,z]. NEWLINE\]NEWLINENEWLINENEWLINE\item (Unidiagonal case) A complete intersection NEWLINE\[NEWLINE \{z^2-f_6(x_i,y)=f_2(x_i)=0\} \subset \mathbb{P}_{(1,1,1,2,3)}[x_1,x_2,x_3,y,z]. NEWLINE\]NEWLINE NEWLINENEWLINE\end{itemize}} Moreover, the assumption on the canonical bundle of \(X\) tell us that the degenerations studied are \textit{Kulikov models}. These models were classified by Kulikov, Friedmann-Morrison and Persson, and are divided into three classes, see [\textit{R. Friedman} and \textit{D. R. Morrison}, Prog. Math. 29, 1--32 (1983; Zbl 0508.14024)]. The author provides a description of the degenerations according to the Kulikov models. In addition, for each type he provides a list of all the possible singularities of \(\phi(X_0)\). Thereby giving a full classification of the possible singular fibres.NEWLINENEWLINEIt is worth to notice that this results proves an analogue of a theorem of \textit{M. Mendes-Lopes} [``The relative canonical algebra for genus three fibrations'', Ph.D. thesis, University of Warwick (1989)] that classifies the canonical ring of degenerate genus two curves. Result which was 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 explicitly construct the relative canonical models of surfaces fibred by genus two curves.