Containments of symbolic powers of ideals of generic points in $\mathbb {P}^3$

Abstract

We show that the Conjecture of Harbourne and Huneke, stating that $I^{(Nr-(N-1))} \subset M^{(r-1)(N-1)}I^{r}$ for ideals of points in $\mathbb {P}^N$, holds for generic (simple) points for $N = 3$. As a result, for such ideals we prove the following bounds, which can be recognized as generalizations of Chudnovsky bounds: $\alpha (I^{(3m-k)}) \geq m\alpha (I)+2m-k$, for any $m \geq 1$ and $k=0,1,2$. Moreover, we obtain lower bounds for the Waldschmidt constant for such ideals.