On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of $${\mathbb{P}^3}$$

Abstract

We consider Cremona transformations of \({\mathbb{P}^3}\) which either stabilize the set of planes passing through a point or they are defined by the maximal minors of a 4 × 3 matrix of linear forms. For a general transformation satisfying one of these two conditions we give an explicit decomposition as a product of elementary links; in other words, we perform the so-called Sarkisov program for (general) transformations belonging to these two classes. Conversely, we show that a Cremona transformation whose decomposition coincides with one of those we have obtained belongs to one of these classes.