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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 ?

PS

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.

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«Does it act on submanifolds?» There is a classical theory of duality for subvarieties of projective space: for example, search for dual curve. –  Mariano Suárez-Alvarez Sep 24 '12 at 19:56

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