On special quadratic birational transformations of a projective space

TL;DR

This paper classifies special quadratic birational transformations of projective spaces into various varieties, revealing interesting base locus manifolds like Severi and homogeneous varieties, with focus on transformations defined by quadratic equations.

Contribution

It provides a classification of quadratic birational transformations into specific varieties, highlighting the structure of their base loci and their relation to known special manifolds.

Findings

Classified quadro-quadric transformations into quadric hypersurfaces.

Classified quadro-cubic transformations into del Pezzo varieties.

Analyzed transformations with base locus dimension up to three.

Abstract

A birational map from a projective space onto a not too much singular projective variety with a single irreducible non-singular base locus scheme (special birational transformation) is a rare enough phenomenon to allow meaningful and concise classification results. We shall concentrate on transformations defined by quadratic equations onto some varieties (especially projective hypersurfaces of small degree), where quite surprisingly the base loci are interesting projective manifolds appearing in other contexts; for example, exceptions for adjunction theory, small degree or small codimensional manifolds, Severi or more generally homogeneous varieties. In particular, we shall classify: quadro-quadric transformations into a quadric hypersurface; quadro-cubic transformations into a del Pezzo variety; transformations whose base locus (scheme) has dimension at most three.