Containments of symbolic powers of ideals of generic points in $\PP^3$

Abstract

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

References in corpus (3)

• Regularity and non-emptyness of linear systems in ${\mathbb{P}}^n$
• Symbolic powers of ideals of generic points in P^3
• Linear systems in P^3 with low degrees and low multiplicities