ABSTRACT. We present a study of certain singular one-parameter subfamilies of Calabi-Yau threefolds realized as anticanonical hypersurfaces or complete intersections in toric varieties. Our attention to these families is motivated by the Doran-Morgan classification of variations of Hodge structure which can underlie families of Calabi-Yau threefolds with h2,1 = 1 over the thrice-punctured sphere. We explore their torically induced fibrations by M-polarized K3 surfaces and use these fibrations to construct an explicit geometric transition between an anticanonical hypersurface and a nef complete intersection through a singular subfamily of hypersurfaces. Moreover, we show that another singular subfamily provides a geometric realization of the missing “14th case” variation of Hodge structure from the Doran-Morgan list.

Introduction

In their paper [DM06], Doran and Morgan give a classification of the possible variations of Hodge structure that can underlie families of Calabi-Yau threefolds with h2,1 = 1 over the thrice-punctured sphere. They find fourteen possibilities. At the time of publication of [DM06], explicit families of Calabi-Yau threefolds realising thirteen of these cases were known and are given in [DM06, Table 1]. The aim of this paper is to give a geometric example which realizes the fourteenth and final case (henceforth known as the 14th case) from their classification, and to study its properties.

By analogy with other examples (see [DM06, Section 4.2]), one might expect that the 14th case variation of Hodge structure should be realized by the mirror of a complete intersection of bidegree Wℙ(2,12) in the weighted projective space (1, 1, 1, 1, 4, 6). However, this ambient space is not Fano, so the Batyrev-Borisov mirror construction cannot be applied to obtain such a mirror family.

Instead, Kreuzer and Sheidegger [KKRS05] suggest working with a slightly different ambient space, given by a non-crepant blow up of WP(1, 1, 1, 1, 4,6). However, the complete intersection Calabi-Yau threefold of bidegree (2,12) in this ambient space has h1,1 = 3, so its mirror will have h2,1 = 3, making it unsuitable as a candidate for the 14th case on the Doran-Morgan list.