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

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- by Marcin Dumnicki PDF
- 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.## References

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## Additional Information

**Marcin Dumnicki**- Affiliation: Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
- MR Author ID: 692599
- Email: Marcin.Dumnicki@im.uj.edu.pl
- Received by editor(s): November 29, 2012
- Received by editor(s) in revised form: May 17, 2013
- Published electronically: October 31, 2014
- Communicated by: Lev Borisov
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**143**(2015), 513-530 - MSC (2010): Primary 14Q10, 13P10
- DOI: https://doi.org/10.1090/S0002-9939-2014-12273-8
- MathSciNet review: 3283641