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Abstract

Conditions characterizing the membership of the ideal of a subvariety ${\mathfrak S}$ arising from (effective) divisors in a product complex space $Y \times X$ are given. For the algebra ${\mathcal O}_Y [V]$ of relative regular functions on an algebraic variety $V$, the strict stability is proved, in the case where $Y$ is a normal space, and the Noether stability is established under a weakened condition. As a consequence (for both general and complete intersections) a global Nullstellensatz is derived for divisors in $Y\times {\mathbb C}^N$, respectively, $Y\times {\mathbb P}^N ({\mathbb C})$. Also obtained are a principal ideal theorem for relative divisors, a generalization of the Gauss decomposition rule, and characterizations of solid pseudospherical harmonics on a semi-Riemann domain. A result towards a more general case is as follows: Let ${\mathfrak D}_j$, $1 \le j\le p$, be principal divisors in $X$ associated to the components of a $q$-weakly normal map $g = (g_1,\ldots,g_p) : X \to {\mathbb C}^p$, and $S := \bigcap {\mathfrak S}_{|{\mathfrak D}_j|}$. Then for any proper slicing $(\varphi,g,D)$ of $\{{\mathfrak D}_j\}_{1\le j\le p}$ (where $D\subset X$ is a relatively compact open subset), there exists an explicitly determined Hilbert exponent ${\mathfrak h}_{_{{\mathfrak D}_1 \cdots {\mathfrak D}_p,D}}$ for the ideal of the subvariety ${\mathfrak S} = Y\times (S\cap D)$.