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Demailly’s Conjecture for m = 2

Establish that for the defining ideal I of a finite set of points X = {P1, …, Ps} in PN over an algebraically closed field, the Waldschmidt constant â(I) satisfies the inequality â(I) ≥ (a(I(2)) + N − 1)/(N + 1).

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Background

The case m = 2 of Demailly’s Conjecture connects â(I) to the initial degree of the second symbolic power I(2). It is proven in several regimes and configurations, including very general points and many general-point cases, but not completely settled for all general point sets.

This paper proves the conjecture for all very general points and for many general-point configurations across dimensions, leaving a small set of cases unresolved in P5.

References

Conjecture 1.2. If Ix is the defining ideal of X = { P1, ... , Ps} C PC, then

a(I(2)) + N-1 â(I) > N +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 1.2, Section 1 (Introduction)