Smoothness of the images of members of a linear system under an endomorphism of the affine plane
type: Item
instance of: scholarly article
title: Smoothness of the images of members of a linear system under an endomorphism of the affine plane
zbMATH Open document ID: 1163.14034
author: Pi. Cassou-Noguès
author: M. Miyanishi
published in: Journal of Pure and Applied Algebra
publication date: 2009-03-10T00:00:00Z
review text: Endomorphisms of the complex plane are studied. Let \(f,g\in \mathbb C [x,y]\) and let \(\varphi=(f,g)\) be an endomorphism of \( \mathbb C^2.\) For \(b\in \mathbb C\) let \(C_b=\{x=b\}.\) The question whether the image \(\varphi (C_b)\) is smooth for a general \(b\) is connected to the Jacobian conjecture. For a pair \(f,g\) and a point \(a\in \mathbb C \) the authors define a special polynomial \(H_a\) and prove that the smoothness of \(\varphi (C_b)\) for general \(b\) is equivalent to the non-vanishing of \(H_a\) for almost all \(a\in \mathbb C.\) Then they introduce a hypersurface \(V\in \mathbb C ^3\) such that its coordinate ring is \(\mathbb C [x,f,g].\) The normalization \(\tilde V\) of \(V\) appears to be a \(\mathbb C ^1\)-fibration with only cyclic singularities. The inclusion \(\mathbb C [x,f,g]\subset \mathbb C [x,y] \) induces a morphism \(\pi:V\to \mathbb C^2.\) It is provend that: - if \(\pi\) is finite, then \(\tilde V\) is isomorphic to \(\mathbb C^2/G,\) where \(G\) is a cyclic group; - if \(\mathbb C (x,f,g)= \mathbb C (x,y) ,\) then \(\tilde V\) is isomorphic to \(\mathbb C^2.\)
reviewed by: Tatiana Bandman
Mathematics Subject Classification ID: 14R10
Mathematics Subject Classification ID: 14R25
zbMATH DE Number: 5527492
zbMATH Keywords: automorphisms of the plane
zbMATH Keywords: \(\mathbb C^1\)-fibrations
zbMATH Keywords: Taylorian resultant
zbMATH Keywords: \(\mathbb Q\)-plane
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.09.002
OpenAlex ID: W2038353129
cites work: Some remarks on the Jacobian question. (Notes by Marius van der Put and William Heinzer, updated by Avinash Sathaye)
cites work: Q3491708
cites work: Q4097032
cites work: Jacobian quotients of polynomial mappings.
cites work: Affine surfaces with \(AK(S)=\mathbb C\).
cites work: The effect of rational maps on polynomial maps
cites work: Q5428578
cites work: Birational endomorphisms of the affine plane
cites work: Local trees in the theory of affine plane curves
cites work: Automorphisms of affine surfaces with \(\mathbb A^1\)-fibrations
cites work: Injectivity on one line
cites work: On flat fibrations by the affine line
cites work: Flat families of affine lines are affine-line bundles
cites work: An inversion formula for two polynomials in two variables
cites work: A chain rule for the resultant of two polynomials
cites work: Lectures on curves on rational and unirational surfaces
cites work: Singularities of normal affine surfaces containing cylinderlike open sets
cites work: Affine pseudo-coverings of algebraic surfaces
cites work: A geometric approach to the Jacobian conjecture in dimension two
cites work: Homology planes with quotient singularities
cites work: A counterexample to the strong real Jacobian conjecture
cites work: Q4055870
cites work: Affine surfaces fibered by affine lines over the projective line
DOI: 10.1016/J.JPAA.2008.09.002
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description: scientific article; zbMATH DE number 5527492