On the Cremona transformations of bidegree (3,3) (Q1384143)

Let \(\mathbb{P}^3\) be three-dimensional projective space over an algebraically closed field \(k\) of characteristic zero. In the paper all birational transformations \(T\) of \(\mathbb{P}^3\) of bidegree \((3,3)\) (this means that equally \(T\) and \(T^{-1}\) have degree \(3\)) are classified into three (not disjoint) classes: (1) the class of determinantal transformations (i.e. components of \(T\) are minors of a \(4\times 3\) matrix the entries of which are linear forms on \(k^4\)), (2) the Jonquières class of transformations (i.e. the strict transformation \(\overline{T^{-1}(L)}\) of a generic line \(L\) is a plane cubic), (3) the ruled class of transformations (i.e. the strict transformation \(\overline{T^{-1}(H)}\) of a generic plane \(H\) is a ruled cubic surface).