Type III contractions and quintic threefolds

Abstract

We study type III contractions of Calabi–Yau threefolds containing a ruled surface over a smooth curve. We discuss the conditions necessary for the image threefold to be smoothable. We describe the change in Hodge numbers caused by this contraction and smoothing deformation. A generalization of a formula for calculating Hodge numbers of hypersurfaces in [Formula: see text] with ordinary double and triple points is presented. We use these results to construct new Calabi–Yau threefolds of Picard rank two arising from a family of quintic threefolds containing a cone.