The minimal free resolution of fat almost complete intersections in ℙ1 x ℙ1

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.