Birational transformations

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

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On birational transformations of pairs in the complex plane

Geometriae Dedicata ◽

10.1007/s10711-008-9316-3 ◽

2008 ◽

Vol 139 (1) ◽

pp. 57-73 ◽

Cited By ~ 3

Author(s):

Jérémy Blanc ◽

Ivan Pan ◽

Thierry Vust

Keyword(s):

Complex Plane ◽

Birational Transformations

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Stability, birational transformations, and the Kähler-Einstein problem

Surveys in Differential Geometry ◽

10.4310/sdg.2012.v17.n1.a5 ◽

2012 ◽

Vol 17 (1) ◽

pp. 203-228 ◽

Cited By ~ 9

Author(s):

S. K. Donaldson

Keyword(s):

Birational Transformations

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Elements Generating a Proper Normal Subgroup of the Cremona Group

International Mathematics Research Notices ◽

10.1093/imrn/rnaa026 ◽

2020 ◽

Author(s):

Serge Cantat ◽

Vincent Guirardel ◽

Anne Lonjou

Keyword(s):

Normal Subgroup ◽

Projective Plane ◽

Infinite Order ◽

Closed Field ◽

Algebraically Closed Field ◽

Cremona Group ◽

Birational Transformations ◽

Proper Normal Subgroup

Abstract Consider an algebraically closed field ${\textbf{k}}$, and let $\textsf{Cr}_2({\textbf{k}})$ be the Cremona group of all birational transformations of the projective plane over ${\textbf{k}}$. We characterize infinite order elements $g\in \textsf{Cr}_2({\textbf{k}})$ having a power $g^n$, $n\neq 0$, generating a proper normal subgroup of $\textsf{Cr}_2({\textbf{k}})$.

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