The minimal free resolution of fat almost complete intersections in $\mathbb{P}^1\times\mathbb{P}^1$

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) sets 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 homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. 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 $m\geq 1.$