On special quadratic birational transformations of a projective space into a hypersurface

TL;DR

This paper classifies special quadratic birational transformations of projective spaces into quadrics, identifying only four cases with base loci related to Severi varieties, focusing on those with smooth connected base loci and quadratic inverses.

Contribution

It provides a complete classification of special quadratic birational maps into quadrics with quadratic inverses, linking them to Severi varieties.

Findings

Only four such transformations exist.

Base loci are general hyperplane sections of Severi varieties.

Identifies conditions for transformations with smooth connected base loci.

Abstract

We study transformations as in the title with emphasis on those having smooth connected base locus, called "special". In particular, we classify all special quadratic birational maps into a quadric hypersurface whose inverse is given by quadratic forms by showing that there are only four examples having general hyperplane sections of Severi varieties as base loci.