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The Minimal Free Resolution of Fat Almost Complete Intersections in ℙ1 × ℙ1
Published online by Cambridge University Press: 20 November 2018
Abstract
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of
$I$. In this paper, we focus on the case where
$I\,=\,{{I}_{X}}$ is an ideal defining an almost complete intersection (ACI) set of points
$X$ in
${{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}$. In particular, we describe a minimal free bigraded resolution of a non-arithmetically Cohen-Macaulay (also non-homogeneous) set
$Z$ of fat points whose support is an ACI, generalizing an earlier result of Cooper et al. for homogeneous sets of triple points. We call
$Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e,
$I_{Z}^{\left( m \right)}\,=\,I_{Z}^{m}$ for any
$m\,\ge \,1$.
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- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2016
References
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