Veneroni maps

Abstract

Veneroni maps are a class of birational transformations of projective spaces. This class contains the classical Cremona transformation of the plane, the cubo-cubic transformation of the space and the quatro-quartic transformation of P 4 . Their common feature is that they are determined by linear systems of forms of degree n vanishing along n + 1 general flats of codimension 2 in P n . They have appeared recently in a work devoted to the so called unexpected hypersurfaces. The purpose of this work is to refresh the collective memory of the mathematical community about these somewhat forgotten transformations and to provide an elementary description of their basic properties given from a modern point of view.

Actual and virtual dimension of codimension 2 general linear subspaces in $\mathbb{P}^n$

“…This map is called the Veneroni map. As it is proved in [5], its inverse is again the Veneroni map defined by the degree n forms vanishing on some general linear subspaces Π ′ 1 , . .…”

“…Denote also by W ij the sets of conditions imposed by intersections Π i ∩ Π j . There are 5 2 = 10 such intersections and those are points in this case, hence each gives one condition. Now each W i should contain the conditions that are already in sets W ij for j = i, hence 4 conditions, whereas one codimension 2 subspace in P 4 gives 7+2 2 = 36 conditions.…”

“…Now we present the construction of Veneroni maps. One can find more details in [5], where authors present basic facts about Veneroni maps with a modern approach. In [7] one can find a thorough analysis of the case of a cubo-cubic transformation in P 3 , which is the special case of the Veneroni map.…”