Special cubic Cremona transformations of $\mathbb{P}^6$ and $\mathbb{P}^7$

TL;DR

This paper completes the classification of special Cremona transformations with three-dimensional base loci, focusing on the case of cubo-quintic transformations of , building on prior classifications of lower-dimensional cases.

Contribution

It provides the classification of special cubo-quintic Cremona transformations of , finalizing the classification for transformations with three-dimensional base loci.

Findings

Classified all special cubo-quintic Cremona transformations of .

Extended the classification of special Cremona transformations to include the three-dimensional base locus case.

Connected previous classifications of lower-dimensional cases to complete the overall picture.

Abstract

A famous result of B. Crauder and S. Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. Furthermore, they also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of $\mathbb{P}^5$; 2) a cubo-quintic transformation of $\mathbb{P}^6$; or 3) a quadro-quintic transformation of $\mathbb{P}^8$. Special Cremona transformations as in case 1) have been classified by L. Ein and N. Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of $\mathbb{P}^8$. The main aim here is to consider the problem of classifying special cubo-quintic Cremona transformations of $\mathbb{P}^6$, concluding the classification of special Cremona transformations whose base locus has dimension three.