Comparing powers and symbolic powers of ideals

By Cristiano Bocci, Brian Harbourne. 2007

235 citations recorded by the source corpus.

We develop tools to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. We obtain results on the structure of the set of pairs $(r,m)$ such that $I^{(m)}\subseteq I^r$. As corollaries, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/711118d18c7b0026c14aeaee4cf0c7a506064df1

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Citations in the corpus

On the containment problem (2016, methodology): “A series of examples was constructed by Bocci and Harbourne in [5].”
Star configurations in Pn (2013, background): “12 is true when c = n = N − 1 and s = n + 2: using ρ(IVn(Pn)) = n(s − n + 1)/s [3], we have ρ(IVn(Pn)) = n(s− n+ 1)/s = 3(N − 1)/(N + 1) = ρ(IVN−1(PN )).”
The Waldschmidt constant for squarefree monomial ideals (2015, background): “Bocci and Harbourne’s result α(I)/α̂(I) ≤ ρ(I) (see [3, Theorem 1.2]) has renewed interest in computing α̂(I).”
Symbolic Powers of Ideals (2015, result): “er generated by yn, which is not in p2n−t for any ﬁxed constant t, as ngets large. Even the results of Theorem 2.19 and 2.20 cannot be improved in an asymptotic sense, as shown by Bocci and Harbourne [BH10a]. All of these results lead to the following question: Question 3.8. Let (R,m,K) be a complet”