Determinantal Quintics and Mirror Symmetry of Reye Congruences

We study a certain family of determinantal quintic hypersurfaces in 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 P 4 × P 4 which have appeared in our previous study of Reye congruences in dimension three. We compactify the two dimensional family over P 2 and reproduce the mirror family to the Reye congruences. We also determine the monodromy of the family over P 2 completely. Our calculation shows an example of the orbifold mirror construction with a trivial orbifold group.