Type III contractions and quintic threefolds. (English) Zbl 1493.14060
Summary: 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 \(\mathbb{P}^4\) 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.
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