Special birational transformations of type (2,1)

TL;DR

This paper classifies special birational transformations of type (2,1) involving nonsingular varieties with Picard number 1, using C^*-actions to extend previous classifications and connect to varieties with nonzero prolongations.

Contribution

It provides a complete classification of special birational transformations of type (2,1), employing C^*-actions to enhance prior results and establish new links to smooth projective varieties.

Findings

Classification of all special birational transformations of type (2,1).

Use of C^*-actions as a key tool in the classification.

Connection established between these transformations and varieties with nonzero prolongations.

Abstract

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