From Demailly's Inequality to Harbourne--Huneke Containments for General Points

Abstract: 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 $\mathbb{P}<sup>N$ 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.