Determinantal Quintics and Mirror Symmetry of Reye Congruences

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.

arXiv comment47 pages, 8 figures; acknowledgement added

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Cited by in corpus (3)

• Double quintic symmetroids, Reye congruences, and their derived equivalence
• On $(2,4)$ complete intersection threefolds that contain an Enriques surface
• A Strange Family of Calabi-Yau 3-folds