On smooth Calabi-Yau threefolds of Picard number two

Abstract

We classify all smooth Calabi-Yau threefolds of Picard number two that have a general hypersurface Cox ring.2010 Mathematics Subject Classification. 14J32. number two with specifying data from distinct items of the table are not isomorphic to each other.Note that the varieties from Theorem 1.1 constitute a finite number of families. For recent general results on boundedness of Calabi-Yau threefolds we refer to [13] as well as [38] for the case of Picard number two.Hypersurfaces in toric Fano varieties form a rich source of examples for Calabi-Yau varieties, e.g. [1,6,7]. Theorem 1.1 comprises several varieties of this type.Remark 1.2. Any Mori dream space X can be embedded into a projective toric variety by choosing a graded presentation of its Cox ring R(X); see [5, Sec. 3.2.5] for details. The following table shows for which varieties X from Theorem 1.1 the presentation R(X) = R g gives rise to an embedding into a (possibly singular) toric Fano variety. Observe that in our situation this simply means µ ∈ Ample(X).