Type III contractions and quintic threefolds

TL;DR

This paper investigates specific contractions in Calabi-Yau threefolds, analyzing their smoothability, Hodge number changes, and constructing new examples of Calabi-Yau threefolds with particular properties.

Contribution

It introduces a generalized formula for Hodge numbers of hypersurfaces with certain singularities and constructs new Calabi-Yau threefolds from quintic threefolds with cones.

Findings

Conditions for smoothability of the contracted threefolds

Change in Hodge numbers due to contractions and deformations

Construction of new Calabi-Yau threefolds with Picard rank two

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 by smoothable. We describe the change in Hodge numbers caused by this contraction and smoothing deformation. A generalization of a fomula 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.