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2013 | 107 | 1 | 1-28
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On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

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Conditions characterizing the membership of the ideal of a subvariety 𝔖 arising from (effective) divisors in a product complex space Y × X are given. For the algebra $𝓞_Y[V]$ 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 $Y × ℂ^N$, respectively, $Y × ℙ^N(ℂ)$. 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 $𝔇_j$, 1 ≤ j ≤ p, be principal divisors in X associated to the components of a q-weakly normal map $g = (g₁,...,g_p) : X → ℂ^p$, and $S := ⋂ 𝔖_{|𝔇_j|}$. Then for any proper slicing (φ,g,D) of ${𝔇_j}_{1≤j≤p}$ (where D ⊂ X is a relatively compact open subset), there exists an explicitly determined Hilbert exponent $𝔥_{𝔇₁ ⋯ 𝔇_p,D}$ for the ideal of the subvariety 𝔖 = Y× (S∩D).
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  • Department of Mathematics and Statistics, Minnesota State University, Mankato, Mankato, MN 56001, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-ap107-1-1
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