On the connectedness of the standard web of Calabi–Yau $3$-folds and small transitions

We supply a detailed proof of the result by P.S. Green and T. Hübsch that all complete intersection Calabi–Yau $3$-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call primitive small transitions and study such subclass. More precisely, given a small projective resolution $\pi : \hat{X} \to X$ of a Calabi–Yau $3$-fold $X$, we show that if the natural closed immersion $\mathrm{Def}(\hat{X}) \hookrightarrow \mathrm{Def}$ is an isomorphism then $X$ has only ODPs as singularities.

Keywords

conifold transition, Calabi–Yau, complete intersection, deformation

2010 Mathematics Subject Classification

14B07, 14B12, 14J30, 14J32, 32Gxx

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Supported by MOST project 103-2115-M-002-002-MY3.

Received 2 September 2015

Accepted 8 February 2018

Published 6 February 2019