On a conjecture of Demailly and new bounds on Waldschmidt constants in ${\mathbb P}^N$

TL;DR

This paper proves Demailly's conjecture for large finite sets of very general points in projective spaces, establishing new lower bounds on Waldschmidt constants, which are important asymptotic invariants in algebraic geometry.

Contribution

It provides the first proof of Demailly's conjecture for certain point sets and introduces improved bounds on Waldschmidt constants.

Findings

Proof of Demailly's conjecture for large finite sets of points

New lower bounds on Waldschmidt constants in projective spaces

Enhanced understanding of asymptotic invariants in algebraic geometry

Abstract

In the present note we prove a conjecture of Demailly for finite sets of sufficiently many very general points in projective spaces. This gives a lower bound on Waldschmidt constants of such sets. Waldschmidt constants are asymptotic invariants of subschemes receiving recently considerable amount of attention.