Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Comparing powers and symbolic powers of ideals

Authors: Cristiano Bocci and Brian Harbourne
Journal: J. Algebraic Geom. 19 (2010), 399-417
Published electronically: August 17, 2009
MathSciNet review: 2629595
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Abstract | References | Additional Information

Abstract: 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 $k[\textbf {P}^N]$ in $N+1$ variables over an arbitrary 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 $\textbf {P}^2$ (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein–Lazarsfeld–Smith [Invent. Math. 144 (2001), pp. 241–252] and Hochster–Huneke [Invent. Math. 147 (2002), pp. 349–369] are optimal for every fixed dimension and codimension.

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Additional Information

Cristiano Bocci
Affiliation: Dipartimento di Scienze Matematiche e Informatiche “R. Magari”, Università degli Studi di Siena, Pian dei mantellini, 44, 53100 Siena, Italy

Brian Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
MR Author ID: 217048

Received by editor(s): February 12, 2008
Published electronically: August 17, 2009
Additional Notes: This reseach was partially supported by Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of Istituto Nazionale di Alta Matematica (Italy) and by the National Security Agency