Special birational transformations of type (2,1)

Abstract

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