Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in $\mathbb{P}^n$
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 \[ \widehat{\alpha}(Z)\geq\frac{\alpha(mZ)+n1}{m+n1}. \] Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor2\geq m1. \] Here we improve their result and show that Demailly's conjecture holds if $Z$ is very general and \[ \lfloor\sqrt[n]{s}\rfloor2\ge \frac{2\varepsilon}{n1}(m1), \] 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 NagataIarrobino conjecture $\widehat{\alpha}(Z)\ge\sqrt[n]{s}$.
 Publication:

arXiv eprints
 Pub Date:
 March 2019
 arXiv:
 arXiv:1903.05824
 Bibcode:
 2019arXiv190305824C
 Keywords:

 Mathematics  Algebraic Geometry;
 14C20