Lower bounds for Waldschmidt constants and Demailly's Conjecture for general and very general points

TL;DR

This work addresses lower bounds for the Waldschmidt constant and Demailly's conjecture for sets of points in projective space. The authors combine Alexander–Hirschowitz bounds, regularity of the second symbolic power, Cremona reductions, and Waldschmidt decompositions to control both $\alpha(I^{(2)})$ and $\widehat{\alpha}(I)$, and they establish containment-based strategies to transfer these bounds into Demailly's inequality. They prove the conjecture for all very general points (in $\mathbb{P}^N$, $N\ge 3$), for at least $2^N$ general points, and for general points in low dimensions ${\mathbb P}^3$, ${\mathbb P}^4$, and most cases in ${\mathbb P}^5$, with a small set of unresolved cases in ${\mathbb P}^5$. The results significantly extend the range of configurations for which Demailly's bound is verified and illuminate the connections between Waldschmidt constants, regularity, and containment problems for symbolic powers.

Abstract

We prove Demailly's Conjecture concerning the lower bound for the Waldschmidt constant in terms of the initial degree of the second symbolic powers for any set of generic points or very general points in $\mathbb{P}^N$. We also discuss the Harbourne-Huneke Containment and the aforementioned Demailly's Conjecture for general points and show the results for sufficiently many general points and general points in projective spaces with low dimensions.

Theorems & Definitions (39)

• Conjecture 1.1
• Conjecture 1.2
• Definition 2.1
• Definition 2.2
• Conjecture 2.3: Demailly
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8: Theorem 4.1 localeffectivity