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

TL;DR

This paper proves Chudnovsky's conjecture and the stable Harbourne-Huneke containment for all configurations of general points in projective spaces, using a new Cremona reduction technique to establish effective bounds.

Contribution

It extends previous results by confirming the conjectures for all cases of general points in any projective space, introducing a Cremona reduction method for bounds.

Findings

Confirmed Chudnovsky's conjecture for all general points in projective spaces.

Established Harbourne-Huneke containment universally for general points.

Developed a Cremona reduction process to estimate Waldschmidt constants.

Abstract

In our previous work with Grifo and H\`a, we showed the stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of sufficiently many general points in $\mathbb{P}^N$. In this paper, we establish the conjectures for all remaining cases, and hence, give the affirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecture for any number of general points in $\mathbb{P}^N$ for all $N$. Our new technique is to develop the Cremona reduction process that provides effective lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective spaces.