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

Moser-Jauslin, Lucy, and Poloni, Pierre-Marie. "Embeddings of a family of Danielewski hypersurfaces and certain $\mathbf{C}^+$-actions on $\mathbf{C}^3$." Annales de l’institut Fourier 56.5 (2006): 1567-1581. <http://eudml.org/doc/10184>.

@article{Moser2006,
abstract = {We consider the family of polynomials in $\mathbf\{C\}[x,y,z]$ of the form $x^2y-z^2-xq(x,z)$. Two such polynomials $P_1$ and $P_2$ are equivalent if there is an automorphism $\varphi ^*$ of $\mathbf\{C\}[x,y,z]$ such that $\varphi ^*(P_1)=P_2$. 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 $c\in \mathbf\{C\}$. There exist polynomials which are algebraically inequivalent but analytically equivalent. There exist polynomials which are algebraically inequivalent but when considered as polynomials in $\mathbf\{C\}[x,y,z,w]$ become equivalent. This last result answers a problem posed in [7]. Finally, we get a complete classification of $\mathbf\{C\}^+$-actions on $\mathbf\{C\}^3$ which are defined by a triangular locally nilpotent derivation of the form $x^2\partial /\partial z+(2z+xq(x,z))\partial /\partial y$.},
affiliation = {Université de Bourgogne Institut de Mathématiques de Bourgogne CNRS–UMR 5584 9, avenue Alain Savary B.P. 47870 21078 Dijon (France); Université de Bourgogne Institut de Mathématiques de Bourgogne CNRS–UMR 5584 9, avenue Alain Savary B.P. 47870 21078 Dijon (France)},
author = {Moser-Jauslin, Lucy, Poloni, Pierre-Marie},
journal = {Annales de l’institut Fourier},
keywords = {equivalence of polynomials; stable equivalence; algebraic embeddings; Danielewski surfaces},
language = {eng},
number = {5},
pages = {1567-1581},
publisher = {Association des Annales de l’institut Fourier},
title = {Embeddings of a family of Danielewski hypersurfaces and certain $\mathbf\{C\}^+$-actions on $\mathbf\{C\}^3$},
url = {http://eudml.org/doc/10184},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Moser-Jauslin, Lucy
AU - Poloni, Pierre-Marie
TI - Embeddings of a family of Danielewski hypersurfaces and certain $\mathbf{C}^+$-actions on $\mathbf{C}^3$
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 5
SP - 1567
EP - 1581
AB - We consider the family of polynomials in $\mathbf{C}[x,y,z]$ of the form $x^2y-z^2-xq(x,z)$. Two such polynomials $P_1$ and $P_2$ are equivalent if there is an automorphism $\varphi ^*$ of $\mathbf{C}[x,y,z]$ such that $\varphi ^*(P_1)=P_2$. 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 $c\in \mathbf{C}$. There exist polynomials which are algebraically inequivalent but analytically equivalent. There exist polynomials which are algebraically inequivalent but when considered as polynomials in $\mathbf{C}[x,y,z,w]$ become equivalent. This last result answers a problem posed in [7]. Finally, we get a complete classification of $\mathbf{C}^+$-actions on $\mathbf{C}^3$ which are defined by a triangular locally nilpotent derivation of the form $x^2\partial /\partial z+(2z+xq(x,z))\partial /\partial y$.
LA - eng
KW - equivalence of polynomials; stable equivalence; algebraic embeddings; Danielewski surfaces
UR - http://eudml.org/doc/10184
ER -