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In our previous work with Grifo and H\`a, we showed the stableHarbourne-Huneke containment and Chudnovsky's conjecture for the defining idealof sufficiently many general points in $\mathbb{P}^N$. In this paper, weestablish the conjectures for all remaining cases, and hence, give theaffirmative answer to Harbourne-Huneke containment and Chudnovsky's conjecturefor any number of general points in $\mathbb{P}^N$ for all $N$. Our newtechnique is to develop the Cremona reduction process that provides effectivelower bounds for the Waldschmidt constant of the defining ideals of genericpoints in projective spaces.

Chudnovsky's Conjecture and the stable Harbourne-Huneke containment for general points