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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Baohua Fu and Jun-Muk Hwang
Journal: J. Algebraic Geom. 27 (2018), 55-89
Published electronically: March 10, 2017
MathSciNet review: 3722690
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Abstract | References | Additional Information

Abstract: 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.

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Additional Information

Baohua Fu
Affiliation: Institute of Mathematics, AMSS, Chinese Academy of Sciences, 55 ZhongGuanCun East Road, Beijing, 100190, People’s Republic of China – and – School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, People’s Republic of China

Jun-Muk Hwang
Affiliation: Korea Institute for Advanced Study, Hoegiro 85, Seoul, 130-722, Republic of Korea
MR Author ID: 362260

Received by editor(s): September 21, 2015
Received by editor(s) in revised form: May 4, 2016
Published electronically: March 10, 2017
Additional Notes: The first author was supported by the National Natural Science Foundation of China (11225106 and 11321101). The second author was supported by the National Researcher Program 2010-0020413 of the NRF
Article copyright: © Copyright 2017 University Press, Inc.