Chia-chi Tung
On Hilbert exponent, Noether stability and global Nullstellensatz
An objective of this work is to find conditions characterizing the membership of the ideal of a subvariety $\mathfrak S$ in a product complex space and give an explicit expression for a Hilbert exponent of $\mathfrak S$. A main result obtained is the following: Assume that $X$, $Y$ are normal complex spaces and $S\subset X$ a subvariety admitting a weakly $q$-flat defining map $g:X\rightarrow\mathbb C^p$. Then for each relatively compact open set $D\subset X$, a Hilbert relation over $Y$ holds for all holomorphic functions on $Y\times D$ vanishing on the subvariety $\mathfrak S=Y\times(S\cap D)$, with an explicitly determined Hilbert exponent $\mathfrak h_{D,S}$. Generalizing results of A. Płoski and P. Tworzewski, the Noether stability of relative regular functions on an algebraic variety is proved (under a weakened Noether condition) and, consequently, a global Nullstellensatz is established for subvarieties lying in $Y\times C^N$, respectively, $Y\times\mathbb P^N(\mathbb C)$. Also obtained are conditions for the ideal of a divisor in a product space to admit a principal generator and characterizations of solid pseudospherical harmonics on a semi-Riemann domain.