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 = Ix is an ideal defining an almost complete intersection (ACI) set of points X in ℙ1 x ℙ1. 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, IZ(m) = IZm for any m ≤ 1.
Favacchio G., Guardo E. (2017). The minimal free resolution of fat almost complete intersections in ℙ1 x ℙ1. CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 69(6), 1274-1291 [10.4153/CJM-2016-040-4].
The minimal free resolution of fat almost complete intersections in ℙ1 x ℙ1
Favacchio G.;
2017-01-01
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 = Ix is an ideal defining an almost complete intersection (ACI) set of points X in ℙ1 x ℙ1. 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, IZ(m) = IZm for any m ≤ 1.
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