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Studying the mirror symmetry of a Calabi-Yau threefold X X of the Reye congruence in P 4 P⁴, we conjecture that X X has a non-trivial Fourier-Mukai partner Y Y. We construct Y Y as the double cover of a determinantal quintic in P 4 P⁴ branched over a curve. We also calculate BPS numbers of both X X and Y Y (and also a related Calabi-Yau complete intersection X ~ 0 X₀) using mirror symmetry.
Hosono et al. (Mon,) studied this question.