# Veneroni maps

## TL;DR

Veneroni maps are a class of birational transformations in projective spaces, including classical Cremona and cubo-cubic transformations, characterized by linear systems vanishing along general flats, with recent relevance to unexpected hypersurfaces.

## Contribution

This work revisits Veneroni maps, providing an elementary and modern description of their properties, and highlights their relevance to recent developments in algebraic geometry.

## Key findings

- Includes classical Cremona, cubo-cubic, and quatro-quartic transformations.
- Characterized by linear systems of degree n vanishing along n+1 flats.
- Connected to the study of unexpected hypersurfaces.

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