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# The Minimal Free Resolution of Fat Almost Complete Intersections in $\mathbb{P}^1\times \mathbb{P}^1$

Published:2016-12-23
Printed: Dec 2017
• Giuseppe Favacchio,
Dipartimento di Matematica e Informatica , Viale A. Doria, 6 - 95100 - Catania, Italy
• Elena Guardo,
Dipartimento di Matematica e Informatica , Viale A. Doria, 6 - 95100 - Catania, Italy
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## 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 that $I=I_X$ 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 $\mathcal Z$ of fat points whose support is an ACI, generalizing a result of S. Cooper et al. for homogeneous sets of triple points. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\geq 1.$
 Keywords: points in $\mathbb{P}^1\times \mathbb{P}^1$, symbolic powers, resolution, arithmetically Cohen-Macaulay
 MSC Classifications: 13C40 - Linkage, complete intersections and determinantal ideals [See also 14M06, 14M10, 14M12] 13F20 - Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 13A15 - Ideals; multiplicative ideal theory 14C20 - Divisors, linear systems, invertible sheaves 14M05 - Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) [See also 13F15, 13F45, 13H10]

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