Determinantal Quintics and Mirror Symmetry of Reye Congruences

2012/08/09 by Shinobu Hosono, Hosono, Shinobu, Hiromichi Takagi +1 · 2 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.AG

paper · pdf · doi:10.48550/arxiv.1208.1813

47 pages, 8 figures; acknowledgement added

openalex publication_date 2012/08/09 · arxiv created 2012/09/05 · arxiv updated 2012/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a certain family of determinantal quintic hypersurfaces in ℙ⁴ 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 ℙ⁴×ℙ⁴ which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over ℙ² and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over ℙ² completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.

Citations

• Picard-Fuchs equations and mirror maps for hypersurfaces
• Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties

Cited by

• A single point as a Calabi-Yau zerofold
• On the generalized toroidal completion of period mappings

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