AI 中文总结
利用Demailly不等式和一般Waldschmidt估计，将无限符号条件简化为单个有限插值条件，证明一般点配置在稠密Zariski开族上对所有r满足Harbourne-Huneke包含关系。
AI 中文摘要
设$I$是代数闭域上射影$N$-空间中$s$个一般点的定义理想。Harbourne和Huneke猜想对每个$r\ge 1$有加强的符号包含关系$$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$。对于一般点，这在低维和小点数情形已知，而在任意维数和任意基数下，最佳一般结果是稳定版本，即对某个稠密Zariski开集上的所有充分大的$r$成立。我们证明Hà和Sivakumar最近对任意有限点集证明Demailly猜想消除了稳定性阈值。关键观察是Bisui和Nguyen的严格一般Waldschmidt估计允许选取单个有限符号水平$m_0$。相应插值数的相等性是Zariski开条件。Demailly不等式随后将这个有限条件传播到每个符号幂的下界，标准正则性准则将这些下界转化为对每个$r$的Harbourne--Huneke包含关系。因此，对每个$N\ge2$和每个$s\ge1$，$\mathbb{P}^N$中$s$点构型的一个稠密Zariski开族同时满足对所有$r\ge 1$的Harbourne--Huneke包含关系。该论证还澄清了早期文献中的一个量词问题：在完整Demailly定理之前，一般纤维上的估计自然要么在非常一般集上给出所有$r$，要么仅对$r\gg 0$给出一个开集。新定理用单个有限插值条件取代了无限多个符号条件。
英文摘要
Let $I$ be the defining ideal of a general set of $s$ points in projective $N$-space over an algebraically closed field. Harbourne and Huneke conjectured the strengthened symbolic containment $$ I^{(Nr)}\subseteq \mathfrak{m}^{r(N-1)}I^r $$ for every $r\ge 1$. For general points this was known in low dimension and for small numbers of points, while in arbitrary dimension and arbitrary cardinality the best general result was the stable version, valid on one dense Zariski-open set for all sufficiently large $r$. We show that the recent proof of Demailly's conjecture for arbitrary finite point sets by Hà and Sivakumar removes the stability threshold. The key observation is that the strict generic Waldschmidt estimate of Bisui and Nguyen allows one to select a single finite symbolic level $m_0$. Equality of the corresponding interpolation number is a Zariski-open condition. Demailly's inequality then propagates this one finite condition to lower bounds for every symbolic power, and a standard regularity criterion converts those bounds into the Harbourne--Huneke containment for every $r$. Consequently, for every $N\ge2$ and every $s\ge1$, one dense Zariski-open family of $s$-point configurations in $\mathbb{P}^N$ satisfies the Harbourne--Huneke containment simultaneously for all $r\ge 1$. The argument also clarifies a quantifier issue in the earlier literature: before the full Demailly theorem, estimates on the generic fibre naturally yielded either all $r$ on a very general set, or one open set only for $r\gg 0$. The new theorem replaces infinitely many symbolic conditions by a single finite interpolation condition.
