# Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in $\mathbb{P}^n$

## TL;DR

This paper advances the understanding of Demailly's conjecture on Waldschmidt constants for very general points in projective space, providing new conditions under which the conjecture holds, especially for large sets of points.

## Contribution

The authors improve existing results by establishing broader conditions for Demailly's conjecture to hold for very general points in projective space.

## Key findings

- Demailly's conjecture verified for all m when s is a perfect n-th power.
- Conjecture holds if s exceeds certain bounds related to n, assuming Nagata-Iarrobino conjecture.
- Enhanced conditions for the conjecture's validity for large s and specific geometric configurations.

## Abstract

Let $Z$ be a finite set of $s$ points in the projective space $\mathbb{P}^n$ over an algebraically closed field $F$. For each positive integer $m$, let $\alpha(mZ)$ denote the smallest degree of nonzero homogeneous polynomials in $F[x_0,\ldots,x_n]$ that vanish to order at least $m$ at every point of $Z$. The Waldschmidt constant $\widehat{\alpha}(Z)$ of $Z$ is defined by the limit \[ \widehat{\alpha}(Z)=\lim_{m \to \infty}\frac{\alpha(mZ)}{m}. \] Demailly conjectured that \[ \widehat{\alpha}(Z)\geq\frac{\alpha(mZ)+n-1}{m+n-1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\geq m-1. \] Here we improve their result and show that Demailly's conjecture holds if $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor-2\ge \frac{2\varepsilon}{n-1}(m-1), \] where $0\le \varepsilon<1$ is the fractional part of $\sqrt[n]{s}$. In particular, for $s$ very general points where $\sqrt[n]{s}\in\mathbb{N}$ (namely $\varepsilon=0$), Demailly's conjecture holds for all $m\in\mathbb{N}$. We also show that Demailly's conjecture holds if $Z$ is very general and \[ s\ge\max\{n+7,2^n\}, \] assuming the Nagata-Iarrobino conjecture $\widehat{\alpha}(Z)\ge\sqrt[n]{s}$.