Lower bounds for Waldschmidt constants and Demailly’s conjecture for general and very general points

Abstract We prove Demailly’s Conjecture concerning the lower bound for the Waldschmidt constant in terms of the initial degree of the second symbolic powers for any set of very general points in $${\mathbb {P}}^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> . We also discuss the Harbourne-Huneke Containment and the aforementioned Demailly’s Conjecture for general points and show the results for sufficiently many general points and general points in projective spaces with low dimensions.