Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points

Abstract

We apply Cremona transformation to provide appropriate lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective space. Applying these results, we establish stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of a set of a small number of general points in P 4 . We also prove stable Harbourne-Huneke containment, and thus, Chudnovsky's conjecture for the defining ideal of s general points in P N in the following cases: (a) : N + 4 s N +2 N where N 5; and (b) : 2 N s 3 N where N = 5, 6, 7, and 8.

Demailly's Conjecture and the containment problem

“…Secondly, we need the proper lower bounds for the Waldschmidt constant of the defining ideal of generic points for proving inequality 2. Using reduction methods from [BN22], we establish reasonable lower bounds for the Waldschmidt constant in Theorem 4.4, which leads to Corollary 4.6. Again, using reduction methods and splitting the points, as in [DSS24, BN22], we establish reasonable lower bounds for the Waldschmidt constant in Theorems 5.1, 5.2, and 5.3, which lead to Corollary 5.4.…”

“…Our proof of Demailly's Conjecture when m = 2 relies on the lower bound for the Waldschmidt constant of a set of generic points and the upper bound of the Castelnouvo-Mumford regularity and the initial degree of the second symbolic power of the defining ideal of the set of generic points. To study the former, we utilize the Cremona reduction process, see for instance, [Dum12,Dum15,BN22], and gluing results from the Waldschmidt decomposition technique given in [DSS24]. For the latter, the celebrated Alexander-Hirschowitz Theorem provides bounds for the regularity value and the initial degree of the second symbolic power.…”

“…Next, to handle α(I), in Theorem 3.2, we split the number of points into two adequate parts to use induction on N , reduction methods and results from [DSS24,BN22] to obtain the proper lower bound for α(I) to prove inequality 1 and thus prove Conjecture 1.2 for very general points.…”

“…We believe that our strategies can be used to prove Demailly's Conjecture 1.1 for higher values of m. Following our method, one can derive similar inequalities as (1) and (2). After that computation of adequate lower bounds for the Waldschmidt constant using methods in [DSS24,BN22] or other methods would lead to the desired conjectural bound 1.1. This paper is outlined as follows.…”