Veneroni Maps

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 $\mathbb{P}^4$. 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 $\mathbb{P}^n$. 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.