\(K3\) fibrations on rigid double octic Calabi-Yau threefolds (Q2809371)

This paper studies \(K3\) fibrations on rigid octic Calabi-Yau threefolds. In particular, the paper considers arrangements of eight planes that give rise to rigid Calabi-Yau threefolds. In total \(14\) examples of rigid Calabi-Yau threefolds are presented. They are obtained as double covers of \(\mathbb P^3\) branched along arrangements of eight planes. In each case, a double line of the arrangements is chosen. A pencil of planes containing the line defines a fibration of the threefold. Resolutions of singularities of a double octic induces a \(K3\) smooth model of a generic fiber, singular fibers corresponding to the plane containing special singular points or lines of the arrangement. The Picard group of the Calabi-Yau threefold is computed and a formula for its rank is presented in relation to the ranks of the Picard group of the generic fiber and of the group spanned by components of the singular fibers.NEWLINENEWLINEThe main result is formulated as follows.NEWLINENEWLINETheorem: There are 14 equations of double covers of \(\mathbb P^3\) branched along arrangements of eight planes which give rise to rigid Calabi-Yau threefolds \(V\). In each case, there is a \(K3\) fibration and the following equality hold: NEWLINE\[NEWLINE\text{{rank}}(\text{{Pic}}(V))=1+\sum_S(r(S)-1) +\text{{rank}}(\text{{Pic}}(S_{\text{gen}})),NEWLINE\]NEWLINE where \(r(S)\) is the number of irreducible components of a fiber \(S\) and \(S_{\text{gen}}\) is a generic fiber which is a \(K3\) surface.