Editing Radon transform (section)

==Radon transform in algebraic geometry==

In [[algebraic geometry]], a Radon transform (also known as the ''Brylinski–Radon transform'') is constructed as follows.

Write

:<math>\mathbf P^d \, \stackrel{p_1} \gets \, H \, \stackrel{p_2}\to \, \mathbf P^{\vee, d}</math>

for the [[incidence relation|universal hyperplane]], i.e., ''H'' consists of pairs (''x'', ''h'') where ''x'' is a point in ''d''-dimensional [[projective space]] <math>\mathbf P^d</math> and ''h'' is a point in the [[dual projective space]] (in other words, ''x'' is a line through the origin in (''d''+1)-dimensional [[affine space]], and ''h'' is a hyperplane in that space) such that ''x'' is contained in ''h''.

Then the Brylinksi–Radon transform is the [[functor]] between appropriate [[derived category|derived categories]] of [[étale sheaf|étale sheaves]]

:<math> \operatorname{Rad} := Rp_{2,*} p_1^* : D(\mathbf P^d) \to D(\mathbf P^{\vee, d}).</math>

The main theorem about this transform is that this transform induces an [[equivalence of categories|equivalence]] of the categories of [[perverse sheaf|perverse sheaves]] on the projective space and its dual projective space, up to constant sheaves.<ref>{{harvtxt|Kiehl|Weissauer|2001|loc=Ch. IV, Cor. 2.4}}</ref>