Comparing powers and symbolic powers of ideals

2007/06/25 by Cristiano Bocci, Bocci, Cristiano, Brian Harbourne +1 · 6 citations
Computer Science · Mathematics · #13F20 #14C20 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.0706.3707

openalex publication_date 2007/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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[\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)⊆ I^r. As corollaries, we show that I² contains I⁽³⁾ whenever S is a finite generic set of points in \bf P² (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.

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