We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of that containment holds for generic determinantal ideals and defining ideals of star configurations.
We apply Cremona transformation to provide appropriate lower bounds for the Waldschmidt constant of the defining ideals of generic points in projective space. Applying these results, we establish stable Harbourne-Huneke containment and Chudnovsky's conjecture for the defining ideal of a set of a small number of general points in P 4 . We also prove stable Harbourne-Huneke containment, and thus, Chudnovsky's conjecture for the defining ideal of s general points in P N in the following cases: (a) : N + 4 s N +2 N where N 5; and (b) : 2 N s 3 N where N = 5, 6, 7, and 8.
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