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We provide homological foundations to establish conjectural homologicalprojective dualities between 1) S^2 P^3 and the double cover of the projective9-space branched along the symmetric determinantal quartic, and 2) S^2 P^4 andthe double cover of the symmetric determinantal quintic in the projective14-space branched along the symmetric determinantal locus of rank at most 3.

Towards Homological Projective duality for S^2 P^3 and S^2 P^4