Demailly's conjecture on Waldschmidt constants for sufficiently many very general points in ℙⁿ

2019/03/14 by Yu-Lin Chang, Chang, Yu-Lin, Shin-Yao Jow +1
Mathematics · #14C20 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C20

paper · pdf · doi:10.48550/arxiv.1903.05824

arxiv created 2019/03/14 · arxiv updated 2019/03/15

Abstract

Let Z be a finite set of s points in the projective space ℙⁿ over an algebraically closed field F. For each positive integer m, let α(mZ) denote the smallest degree of nonzero homogeneous polynomials in F[x₀,…,x_n] that vanish to order at least m at every point of Z. The Waldschmidt constant \widehatα(Z) of Z is defined by the limit \widehatα(Z)=lim_{m → ∞}(α(mZ))/(m). Demailly conjectured that \widehatα(Z)≥(α(mZ)+n-1)/(m+n-1). Recently, Malara, Szemberg, and Szpond established Demailly's conjecture when Z is very general and \lfloor√[n]s\rfloor-2≥ m-1. Here we improve their result and show that Demailly's conjecture holds if Z is very general and \lfloor√[n]s\rfloor-2≥ (2ε)/(n-1)(m-1), where 0≤ ε<1 is the fractional part of √[n]s. In particular, for s very general points where √[n]s∈ℕ (namely ε=0), Demailly's conjecture holds for all m∈ℕ. We also show that Demailly's conjecture holds if Z is very general and s≥max\n+7,2ⁿ\, assuming the Nagata-Iarrobino conjecture \widehatα(Z)≥√[n]s.

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