On Halphen’s Theorem and some generalizations

Lins Neto, Alcides. "On Halphen’s Theorem and some generalizations." Annales de l’institut Fourier 56.6 (2006): 1947-1982. <http://eudml.org/doc/10195>.

@article{LinsNeto2006,
abstract = {Let $M^n$ be a germ at $0\in \mathbb\{C\}^\{m\}$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi \colon (\mathbb\{C\}^n,0)\rightarrow (M,0)$ such that $\phi ^\{-1\}(0)=\lbrace 0\rbrace $ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,\ldots ,F_k)$, that is there exists a linear holomorphic vector field $X$ on $\mathbb\{C\}^\{m\}$, with eigenvalues $\lambda _1,\ldots ,\lambda _\{m\}\in \mathbb\{Q\}_+$ such that $X(F^T)= U\cdotpF^T$, where $U$ is a $k\times k$ matrix with entries in $\mathcal\{O\}_\{m\}$. We prove that if there exists a germ of holomorphic map $\phi $ as above and $\dim _\{\mathbb\{C\}\}(\{\rm sing\} (M))\le n-2$, then $\lambda _1+\cdots +\lambda _\{m\}&gt; \{\rm Re\}(\{\rm tr\}(U)(0))$. In Theorem 2 we answer the question completely when $n=2$, $k=1$ and $0$ is an isolated singularity of $M$. In Theorem 3 we prove that, if there exists a map as above, $k=1$ and $\dim _\{\mathbb\{C\}\}(\{\rm sing\} (M))\le n-2$, then $\dim _\{\mathbb\{C\}\}(\{\rm sing\}(M))= n-2$. We observe that Theorems 1 and 2 are generalizations of some results due to Halphen.},
affiliation = {Instituto de Matemática Pura e Aplicada Estrada Dona Castorina, 110 Horto, Rio de Janeiro (Brasil)},
author = {Lins Neto, Alcides},
journal = {Annales de l’institut Fourier},
keywords = {Halphen’s theorem; quasi-homomogeneous; complete intersection; Halphen's theorem},
language = {eng},
number = {6},
pages = {1947-1982},
publisher = {Association des Annales de l’institut Fourier},
title = {On Halphen’s Theorem and some generalizations},
url = {http://eudml.org/doc/10195},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Lins Neto, Alcides
TI - On Halphen’s Theorem and some generalizations
JO - Annales de l’institut Fourier
PY - 2006
PB - Association des Annales de l’institut Fourier
VL - 56
IS - 6
SP - 1947
EP - 1982
AB - Let $M^n$ be a germ at $0\in \mathbb{C}^{m}$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi \colon (\mathbb{C}^n,0)\rightarrow (M,0)$ such that $\phi ^{-1}(0)=\lbrace 0\rbrace $ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,\ldots ,F_k)$, that is there exists a linear holomorphic vector field $X$ on $\mathbb{C}^{m}$, with eigenvalues $\lambda _1,\ldots ,\lambda _{m}\in \mathbb{Q}_+$ such that $X(F^T)= U\cdotpF^T$, where $U$ is a $k\times k$ matrix with entries in $\mathcal{O}_{m}$. We prove that if there exists a germ of holomorphic map $\phi $ as above and $\dim _{\mathbb{C}}({\rm sing} (M))\le n-2$, then $\lambda _1+\cdots +\lambda _{m}&gt; {\rm Re}({\rm tr}(U)(0))$. In Theorem 2 we answer the question completely when $n=2$, $k=1$ and $0$ is an isolated singularity of $M$. In Theorem 3 we prove that, if there exists a map as above, $k=1$ and $\dim _{\mathbb{C}}({\rm sing} (M))\le n-2$, then $\dim _{\mathbb{C}}({\rm sing}(M))= n-2$. We observe that Theorems 1 and 2 are generalizations of some results due to Halphen.
LA - eng
KW - Halphen’s theorem; quasi-homomogeneous; complete intersection; Halphen's theorem
UR - http://eudml.org/doc/10195
ER -