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arXiv · 1208.1813

Determinantal Quintics and Mirror Symmetry of Reye Congruences

Shinobu Hosono·Hiromichi Takagi
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Abstract

We study a certain family of determinantal quintic hypersurfaces in $\mathbb{P}^{4}$ whose singularities are similar to the well-studied Barth-Nieto quintic. Smooth Calabi-Yau threefolds with Hodge numbers $(h^{1,1},h^{2,1})=(52,2)$ are obtained by taking crepant resolutions of the singularities. It turns out that these smooth Calabi-Yau threefolds are in a two dimensional mirror family to the complete intersection Calabi-Yau threefolds in $\mathbb{P}^{4}\times\mathbb{P}^{4}$ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over $\mathbb{P}^{2}$ and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over $\mathbb{P}^{2}$ completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.

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Shinobu Hosono, Hiromichi Takagi. 2012-08-09. Determinantal Quintics and Mirror Symmetry of Reye Congruences. https://arxiv.org/abs/1208.1813

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