  We provide homological foundations to establish conjectural homological
projective dualities between 1) S^2 P^3 and the double cover of the projective
9-space branched along the symmetric determinantal quartic, and 2) S^2 P^4 and
the double cover of the symmetric determinantal quintic in the projective
14-space branched along the symmetric determinantal locus of rank at most 3.
