Consider vector space V and its dual V^* then to any line subspace in V one can correspond its kernel in V^* which is hyperplane. Projective duality states that this correspondence satisfies many remarkable properties.
Question Is there some modern point of view on projective duality ? Does it act somehow on submanifolds of P(V) ? Sending to some submanifolds of P(V^ * ) ?
Something like Fourier-Mukai transform or whatever ?
Fourier transform sends D-modules on V to D-modules on V^ * may it is somehow related ?
One motivation comes from the question.
Also I have the following example for C^3
the affine lines (x,y, *) are mapped to points (x,y, 0 )
the points (0, 0, z) are mapped to 2-dimensional subspaces (*, *, z) .
Can one put this in some framework of "projective duality" ?
There are papers by A. Kuznetsov e.g. http://arxiv.org/abs/math/0507292 "Homological Projective Duality" it might be the answer, but at the moment I am not enough familiar to be sure that it is what I need.