  In this paper, we investigate containment statements between symbolic and
ordinary powers and bounds on the Waldschmidt constant of defining ideals of
points in projective spaces. We establish the stable Harbourne conjecture for
the defining ideal of a general set of points. We also prove Chudnovsky's
Conjecture and the stable version of the Harbourne--Huneke containment
conjectures for a general set of sufficiently many points.

Commutative Algebra;Algebraic Geometry