On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz

Chia-chi Tung

Annales Polonici Mathematici, Tome 107 (2013), p. 1-28 / Harvested from The Polish Digital Mathematics Library

Résumé

Conditions characterizing the membership of the ideal of a subvariety arising from (effective) divisors in a product complex space Y × X are given. For the algebra $_Y[V]$ 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 $_j$, 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 $S := ⋂ _{|_j|}$. Then for any proper slicing (φ,g,D) of ${_j}_{1≤j≤p}$ (where D ⊂ X is a relatively compact open subset), there exists an explicitly determined Hilbert exponent $_{₁ ⋯ _p,D}$ for the ideal of the subvariety = Y× (S∩D).

Détail

Publié le : 2013-01-01

Zbl 1266.32011

EUDML-ID : urn:eudml:doc:280375