Parametrizations of analytic varieties

Abstract:

Let V be an analytic subvariety of an open subset $\Omega$ of ${{\text {C}}^n}$ of pure dimension r; for any $p \in V$, there exists an $n - r$ dim plane T such that ${\pi _T}:V \to {{\text {C}}^r}$, the projection along T to ${{\text {C}}^r}$, is a branched covering of finite sheeting order $\mu (V,p,T)$ in some neighborhood of V about p. ${\pi _T}$ is called a global parametrization of V if ${\pi _T}$ has all discrete fibers, e.g. ${\dim _p}V \cap (T + p) = 0$ for all $p \in V$. Theorem. $B = \{ (p,T) \in V \times G(n - r,n)|{\dim _p}V \cap (T + p) > 0\}$ is an analytic set. If ${\pi _2}:V \times G \to G$ is the natural projection, then ${\pi _2}(B)$ is a negligible set in G. Theorem. $\{ (p,T) \in V \times G|\mu (V,p,T) \geq k\}$ is an analytic set. For each $p \in V$, there is a least $\mu (V,p)$ and greatest $m(V,p)$ sheeting multiplicity over all $T \in G$. If $\Omega$ is Stein, V is the locus of finitely many holomorphic functions but its ideal in $\mathcal {O}(\Omega )$ is not necessarily finitely generated. Theorem. If $\mu (V,p)$ is bounded on V, then its ideal is finitely generated.