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

Chia-chi Tung. "On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz." Annales Polonici Mathematici 107.1 (2013): 1-28. <http://eudml.org/doc/280375>.

@article{Chia2013,
abstract = {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).},
author = {Chia-chi Tung},
journal = {Annales Polonici Mathematici},
keywords = {Noether stability; Hilbert number; Hilbert exponent; mapping multiplicity; intersection number},
language = {eng},
number = {1},
pages = {1-28},
title = {On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz},
url = {http://eudml.org/doc/280375},
volume = {107},
year = {2013},
}

TY - JOUR
AU - Chia-chi Tung
TI - On Noether and strict stability, Hilbert exponent, and relative Nullstellensatz
JO - Annales Polonici Mathematici
PY - 2013
VL - 107
IS - 1
SP - 1
EP - 28
AB - 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).
LA - eng
KW - Noether stability; Hilbert number; Hilbert exponent; mapping multiplicity; intersection number
UR - http://eudml.org/doc/280375
ER -