# What is projective duality from modern point of view ? (correspondence ? Fourier on D-mod ? Aut of D(Coh) ?)

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 $\mathbb{C}^3$

• 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
There is quite a lot on projective duality in Gelfand-Kapranov-Zelevinsky, Discriminants, Resultants, and Multidimensional determinants. – Todd Trimble Oct 16 '15 at 11:51