On Cremona transformations and quadratic complexes
type: Item
instance of: scholarly article
title: On Cremona transformations and quadratic complexes
zbMATH Open document ID: 1184.14024
author: Dan Avritzer
author: Iván Pan
author: Gérardo Gonzalez-Sprinberg
publication date: 2009-10-26T00:00:00Z
review text: A cubo-cubic transformation is a birational map \(\phi : \mathbb{P}^3 \dasharrow \mathbb{P}^3\) of degree 3. According to the classification by H. Hudson, there are three kinds of them: Determinantal, de jonquiéres and ruled. The paper under review gives a new construction of some cubo-cubic transformations by using quadratic line complexes. It is shown that any cubo-cubic transformation \(\phi\) whose base locus contains a smooth quintic curve of genus 2 is constructed in the following way. Let \(X=Q_1 \cap Q_2 \subset \mathbb{P}^5\) a smooth complete intersection of two smooth quadric hypersurfaces. Let \(L_1, L_2 \subset X\) two distinct lines and \(E \subset \mathbb{P}^5\) a 3-plane disjoint from the two lines. Let \(\pi_i : X \dasharrow E\), \(i=1,2\), be the projection from the lines \(L_1\) and \(L_2\) to \(E\cong \mathbb{P}^3\), respectively. The authors show that \(\pi_i\) are birational and hence \(\phi=\pi_1\pi_2^{-1}\) gives a birational automorphism of \(\mathbb{P}^3\) which is cubo-cubic. Moreover, if \(L_1\cap L_2 = \emptyset\), then \(\phi\) is determinantal. Otherwise it is of de jonquiére type.
reviewed by: Nikolaos Tziolas
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14N15
zbMATH DE Number: 5620600
zbMATH Keywords: Cremona transformations
zbMATH Keywords: cubo-cubic transformations
MaRDI profile type: Publication
full work available at URL: s12215-008-0026-3
OpenAlex ID: W2022050837
cites work: Moduli spaces of quadratic complexes and their singular surfaces
cites work: The hilbert scheme component of the intersection of two quadrics
cites work: Q4838936
cites work: Cremona Transformations with Smooth Irreducible Fundamental Locus
cites work: Cremona Transformations and Hartshorne's Conjecture
cites work: Some Special Cremona Transformations
cites work: Q3328651
cites work: On the monomial birational maps of the projective space
cites work: On characteristic classes of determinantal Cremona transformations
cites work: Q4195061
cites work: Q3998727
cites work: Introduction to the Mori program
cites work: Threefolds whose canonical bundles are not numerically effective
cites work: Quadratic complexes. II
cites work: Moduli of vector bundles on a compact Riemann surface
cites work: On the Cremona transformations of bidegree (3,3)
cites work: Les transformations de Cremona stellaires. (The stellar Cremona transformations)
cites work: Cremona transformations and special double structures
cites work: On birational maps and Jacobian matrices.
cites work: Liaison des variétés algébriques. I
cites work: Q5801760
cites work: The T 2,4 of S 6 Defined by a Rational Surface 3 F 8
DOI: 10.1007/S12215-008-0026-3
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5620600