Comparing powers and symbolic powers of ideals

Details Comparing powers and symbolic powers of ideals Comparing powers and symbolic powers of ideals

2007-06-25

arxiv.org/abs/0706.3707v2

Mathematics

Algebraic Geometry

15 pages, to appear, Journal of Algebraic Geometry; version 2: minor expositional changes, typos fixed, and references and Boc

Scientific paper

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.

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