Preprint A265/2003
On Halphen's theorem and some generalizations
Alcides Lins Neto
Keywords: quasi-homogeneous | analytic set | parametrization
\abstract{ Let $M^n$ be a germ at $0\in \Bbb{\C}^{m}$ 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 there exists a germ of holomorphic map $\phi \colon (\Bbb{\C}^n,0)\to (M,0)$ such that $\phi^{-1}(0)=\{0\}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,...,F_k)$, that is there exists a linear holomorphic vector field $X$ on $\C^{m}$, with eigenvalues $\lambda_1,...,\lambda_{m}\in \Q_+$ such that $X(F^T)= U.F^T$, where $U$ is a $k\times k$ matrix with entries in $\Cal{O}_{m}$. We prove that if there exists a germ of holomorphic map $\phi$ as above and $\dim_{\Bbb{\C}}(sing(M))\le n-2$ then $\lambda_1+...+\lambda_{m}>Re(tr(U)(0))$. 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 $\dim_{\Bbb{\C}}(sing(M))\le n-2$, then $\dim_{\Bbb{\C}}(sing(M))= n-2$. We observe that Theorems 1 and 2 are generalizations of some results due to Halphen .}