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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Parametrizations of analytic varieties

Author: Joseph Becker
Journal: Trans. Amer. Math. Soc. 183 (1973), 265-292
MSC: Primary 32B10
MathSciNet review: 0344513
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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)\vert{\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\vert\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.

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Keywords: Bounded multiplicity, finitely generated ideal
Article copyright: © Copyright 1973 American Mathematical Society