Mirror symmetry and projective geometry of Reye congruences I

Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in ${\text{mP}}^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in ${\text{mP}}^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection ${\tilde{X}}_0$) using mirror symmetry.