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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

Giuseppe Favacchio  and
Elena Guardo
Giuseppe Favacchio
Affiliation:
Dipartimento di Matematica e Informatica, Viale A. Doria, 6 - 95100 - Catania, Italy e-mail: [email protected], [email protected]
Elena Guardo
Affiliation:
Dipartimento di Matematica e Informatica, Viale A. Doria, 6 - 95100 - Catania, Italy e-mail: [email protected], [email protected]
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Abstract

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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 $m\,\ge \,1$.

Keywords

13C4013F2013A1514C2014M05points in ℙ1 × ℙ1symbolic powersresolutionarithmetically Cohen-Macaulay
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 6 , 01 December 2017 , pp. 1274 - 1291
Copyright
Copyright © Canadian Mathematical Society 2016

References

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