Demailly’s Conjecture (general m) on Waldschmidt constants

Establish that for the defining ideal I of a finite set of points X = {P1, …, Ps} in PN over an algebraically closed field, and for every integer m ≥ 1, the Waldschmidt constant â(I) satisfies the inequality â(I) ≥ (a(I(m)) + N − 1)/(m + N − 1).

Background

The Waldschmidt constant â(I) measures asymptotic lower bounds for the degrees of forms in symbolic powers I(m) of the defining ideal of a finite set of points in projective space PN. Demailly proposed a uniform lower bound relating â(I) to the initial degree a(I(m)) of I(m) across all m.

This inequality is known in PN for N = 2 and for certain special configurations and ranges in higher dimensions, but remains unproven in full generality. The paper develops techniques via Hilbert functions, Cremona reductions, and containment strategies that prove many cases, yet the conjecture in its most general form is still open.

References

Conjecture 1.1. [Dem82] If Ix is the defining ideal of X = { P1, ... , Ps} C PC, then â(Ix) ≥ a(I) (m) + N - 1 , m+N-1 for all m > 1.

— Lower bounds for Waldschmidt constants and Demailly's Conjecture for general and very general points (2401.11297 - Bisui et al., 20 Jan 2024) in Conjecture 1.1, Section 1 (Introduction)