The Minimal Free Resolution of Fat Almost Complete Intersections in P1 × P1

Favacchio, Giuseppe ; Guardo, Elena

Canadian journal of mathematics, Tome 69 (2017) no. 6, pp. 1274-1291

Résumé

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 $m\,\ge \,1$ .

Détail

DOI : 10.4153/CJM-2016-040-4

Mots-clés : 13C40, 13F20, 13A15, 14C20, 14M05, points in P1 × P1, symbolic powers, resolution, arithmetically Cohen-Macaulay

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