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

Lucy Moser-Jauslin^{[1]}; Pierre-Marie Poloni^{[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 (2006)

- Volume: 56, Issue: 5, page 1567-1581
- ISSN: 0373-0956

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topMoser-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 -

## References

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