Embeddings of a family  of Danielewski hypersurfaces  and certain $\mathbf{C}^+$-actions on $\mathbf{C}^3$

Moser-Jauslin, Lucy; Poloni, Pierre-Marie

doi:10.5802/aif.2220

Embeddings of a family of Danielewski hypersurfaces and certain

C^{+}

-actions on

C^{3}

[Plongements d’une famille d’hypersurfaces de Danielewski et certaines actions de

C^{+}

sur

C^{3}

]

Moser-Jauslin, Lucy ¹ ; Poloni, Pierre-Marie ¹

¹ Université de Bourgogne Institut de Mathématiques de Bourgogne CNRS–UMR 5584 9, avenue Alain Savary B.P. 47870 21078 Dijon (France)

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1567-1581.

Résumé
Abstract

Nous considérons la famille de polynômes de $C [x, y, z]$ de la forme $x^{2} y - z^{2} - x q (x, z) .$ Deux polynômes $P_{1}$ et $P_{2}$ sont dits équivalents s’il existe un automorphisme $ϕ *$ de $C [x, y, z]$ tel que $ϕ * (P_{1}) = P_{2} .$ Nous donnons une classification complète des classes d’équivalence de ces polynômes dans les catégories algébrique et analytique. Nous en déduisons les résultats suivants. Il existe des exemples explicites de polynômes non équivalents $P_{1}$ et $P_{2}$ tels que l’ensemble des zéros de $P_{1} + c$ est isomorphe à l’ensemble des zéros de $P_{2} + c$ pour tout $c \in C$ . Il existe des polynômes analytiquement équivalents qui ne le sont pas algébriquement. Il existe des polynômes algébriquement non équivalents mais qui, vus comme des polynômes de $C [x, y, z, w]$ , le deviennent. Ce dernier résultat répond à un problème posé dans [7]. Finalement, nous obtenons une classification complète des actions de $C^{+}$ sur $C^{3}$ définies par une dérivation triangulaire de la forme $x^{2} \partial / \partial z + (2 z + x q (x, z)) \partial / \partial y .$

We consider the family of polynomials in $C [x, y, z]$ of the form $x^{2} y - z^{2} - x q (x, z)$ . Two such polynomials $P_{1}$ and $P_{2}$ are equivalent if there is an automorphism $ϕ^{*}$ of $C [x, y, z]$ such that $ϕ^{*} (P_{1}) = P_{2}$ . We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials $P_{1}$ and $P_{2}$ such that the zero set of $P_{1} + c$ is isomorphic to the zero set of $P_{2} + c$ for all $c \in C$ . There exist polynomials which are algebraically inequivalent but analytically equivalent. There exist polynomials which are algebraically inequivalent but when considered as polynomials in $C [x, y, z, w]$ become equivalent. This last result answers a problem posed in [7]. Finally, we get a complete classification of $C^{+}$ -actions on $C^{3}$ which are defined by a triangular locally nilpotent derivation of the form $x^{2} \partial / \partial z + (2 z + x q (x, z)) \partial / \partial y$ .

MR Zbl

DOI : 10.5802/aif.2220

Classification : 14R10, 14R05, 14L30
Keywords: equivalence of polynomials, stable equivalence, algebraic embeddings, Danielewski surfaces.
Mot clés : polynômes équivalents, équivalence stable, plongements algébriques, surfaces de Danielewski.

Affiliations des auteurs :

Moser-Jauslin, Lucy ¹ ; Poloni, Pierre-Marie ¹

¹ Université de Bourgogne Institut de Mathématiques de Bourgogne CNRS–UMR 5584 9, avenue Alain Savary B.P. 47870 21078 Dijon (France)

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     title = {Embeddings of a family  of {Danielewski} hypersurfaces  and certain $\mathbf{C}^+$-actions on $\mathbf{C}^3$},
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     pages = {1567--1581},
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Moser-Jauslin, Lucy; Poloni, Pierre-Marie. Embeddings of a family  of Danielewski hypersurfaces  and certain $\mathbf{C}^+$-actions on $\mathbf{C}^3$. Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1567-1581. doi : 10.5802/aif.2220. https://www.numdam.org/articles/10.5802/aif.2220/

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Cité par 11 documents. Sources : Crossref, zbMATH