Duality between S^2 P^4 and the Double Quintic Symmetroid

We obtain homological properties of the second symmetric product of P^4 and the double cover of the symmetric determinantal quintic hypersurface in P^{14} (the double quintic symmetroids), which indicate the homological projective duality between (suitable noncommutative resolutions of) them. Among other things, we construct their good desingularizations and also (dual) Lefschetz collections in the derived categories of the desingularizations. These are expected to give (dual) Lefschetz decompositions of suitable noncommutative resolutions. The desingularization of the double quintic symmetroids also contains its interesting birational geometries.