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


Author: Marcin Dumnicki
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
Published electronically: October 31, 2014
MathSciNet review: 3283641
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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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Marcin Dumnicki
Affiliation: Institute of Mathematics, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
Email: Marcin.Dumnicki@im.uj.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-2014-12273-8
Keywords: Symbolic powers, fat points.
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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.