On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of \(\mathbb P^{3}\)
type: Item
instance of: scholarly article
title: On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of \(\mathbb P^{3}\)
zbMATH Open document ID: 1270.14005
published in: Beiträge zur Algebra und Geometrie
publication date: 2012-02-29T00:00:00Z
review text: In his article [J. Algebr. Geom. 4, No. 2, 223--254, appendix 248--254 (1995; Zbl 0866.14007)], \textit{A. Corti} gave a rigorous proof of the Sarkisov program for threefolds, proving that every birational map between two Mori fiber spaces (of dimension three) factorizes into a product of four types of elementary links. To any such a map \(\Phi: X\dashrightarrow X'\), one associates a suitable triple of rational numbers \((\mu, \lambda, \epsilon)\), called the Sarkisov degree of \(\Phi\). It is possible to construct an elementary link \(\Phi':X\dashrightarrow X_1\) such that the Sarkisov degree of \(\Phi\circ(\Phi')^{-1}\) is strictly less than the Sarkisov degree of \(\Phi\) (with respect to the lexicographic order), so to start an induction process. Such a \(\Phi'\) is said to untwist \(\Phi\). In the article under review, the author studies this situation for three types of Cremona transformations of \(\mathbb P^3\): general de Jonquières transformations, both of degree \(2\) and of degree greater than \(2\), and special cubo-cubic transformations. In all the three cases he gets the precise form of the elementary links appearing in the Sarkisov decomposition, and proves that this form provides a characterization of the three types of Cremona transformations considered.
reviewed by: Emilia Mezzetti
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 6010045
zbMATH Keywords: Cremona transformations
zbMATH Keywords: Sarkisov program
author: Iván Pan
MaRDI profile type: Publication
full work available at URL: s13366-011-0054-6
OpenAlex ID: W2082354463
cites work: Geometry of the plane Cremona maps
cites work: Morphisms of projective varieties from the viewpoint of minimal model theory
cites work: Q4838936
cites work: Cremona Transformations with Smooth Irreducible Fundamental Locus
cites work: Some Special Cremona Transformations
cites work: Q4143433
cites work: Q4650859
cites work: The cubo-cubic transformation of \({\mathbb P}^ 3\) is very special
cites work: Divisorial contractions in dimension three which contract divisors to smooth points
cites work: Q4212603
cites work: Classification of Three-Dimensional Flips
cites work: Q4828540
cites work: Introduction to the Mori program
cites work: Threefolds whose canonical bundles are not numerically effective
cites work: On the Cremona transformations of bidegree (3,3)
cites work: Une remarque sur la g�n�ration du groupe de Cremona
cites work: Les transformations de Cremona stellaires. (The stellar Cremona transformations)
cites work: Cremona transformations and special double structures
cites work: Liaison des variétés algébriques. I
cites work: Q5801760
cites work: Elementary birational maps between Mori toric fiber 3-spaces
DOI: 10.1007/S13366-011-0054-6
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description: scientific article; zbMATH DE number 6010045