Mirror symmetry and projective geometry of Reye congruences I

Studying the mirror symmetry of a Calabi-Yau threefold X X of the Reye congruence in P 4 P^4 , 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 P^4 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 X_0 ) using mirror symmetry.