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

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 P1XP1. 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 Corollary4.6 given in [5] 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^{(m)}= I_Z^m for any $mgeq1$.