Containments of symbolic powers of ideals of generic points in ℙ³

Dumnicki, Marcin

doi:10.1090/S0002-9939-2014-12273-8

Containments of symbolic powers of ideals of generic points in $\mathbb {P}^3$
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Proc. Amer. Math. Soc. 143 (2015), 513-530 Request permission

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.

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