  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.
