LAUSR

Abstract Let Z be a finite set of s points in the projective space P n over an algebraically closed field F. For each positive integer m, let α ( m Z ) denote the smallest degree of nonzero homogeneous polynomials in F [ x 0 , … , x n ] that vanish to order at least m at every point of Z. The Waldschmidt constant α ˆ ( Z ) of Z is defined by the limit α ˆ ( Z ) = lim m → ∞ ⁡ α ( m Z ) m . Demailly conjectured that α ˆ ( Z ) ≥ α ( m Z ) + n − 1 m + n − 1 . Malara, Szemberg, and Szpond [7] established Demailly's conjecture when Z is very general and s ≥ ( m + 1 ) n . Here we improve their result and show that Demailly's conjecture holds if Z is very general and ⌊ s n ⌋ ≥ 2 e n − 1 ( m − 1 ) + 2 , where 0 ≤ e 1 is the fractional part of s n . In particular, for s very general points where s n ∈ N (namely e = 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 , 2 n } , assuming the Nagata-Iarrobino conjecture α ˆ ( Z ) ≥ s n .