Special birational transformations of type $(2,1)$

A birational transformation $\Phi : \mathbb {P}^n \dasharrow Z \subset \mathbb {P}^N,$ where $Z \subset \mathbb {P}^N$ is a nonsingular variety of Picard number 1, is called a special birational transformation of type $(a,b)$ if $\Phi$ is given by a linear system of degree $a$, its inverse $\Phi ^{-1}$ is given by a linear system of degree $b$ and the base locus $S \subset \mathbb {P}^n$ of $\Phi$ is irreducible and nonsingular. In this paper, we classify special birational transformations of type $(2,1)$. In addition to previous works by Alzati-Sierra and Russo on this topic, our proof employs natural $\mathbb {C}^*$-actions on $Z$ in a crucial way. These $\mathbb {C}^*$-actions also relate our result to the prolongation problem studied in our previous work.