Demailly's Conjecture and the Containment Problem

TL;DR

This paper explores Demailly's Conjecture related to bounds on Waldschmidt constants, examines a containment problem between symbolic and ordinary powers, and proves its validity for specific algebraic ideals.

Contribution

It extends the understanding of Demailly's Conjecture and confirms a key containment for generic determinantal ideals and star configurations.

Findings

Demailly's Conjecture holds for general sets of points with many points.

A specific containment between symbolic and ordinary powers is proven for certain ideals.

The results support the conjectured bounds and containments in algebraic geometry.

Abstract

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective spaces. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.