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

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.