1. CJM 2016 (vol 69 pp. 1274)
 Favacchio, Giuseppe; Guardo, Elena

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) 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 CohenMacaulay (also non homogeneous) set $\mathcal
Z$ of fat
points whose support is an ACI, generalizing
a result of S. Cooper et al.
for homogeneous sets of triple points. 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 $m\geq 1.$
Keywords:points in $\mathbb{P}^1\times \mathbb{P}^1$, symbolic powers, resolution, arithmetically CohenMacaulay Categories:13C40, 13F20, 13A15, 14C20, 14M05 

2. CJM 2004 (vol 56 pp. 716)
 Guardo, Elena; Van Tuyl, Adam

Fat Points in $\mathbb{P}^1 \times \mathbb{P}^1$ and Their Hilbert Functions
We study the Hilbert functions of fat points in $\popo$.
If $Z \subseteq \popo$ is an arbitrary fat point scheme, then
it can be shown that for every $i$ and $j$ the values of the Hilbert
function $_{Z}(l,j)$ and $H_{Z}(i,l)$ eventually become constant for
$l \gg 0$. We show how to determine these eventual values
by using only the multiplicities of the points, and the
relative positions of the points in $\popo$. This enables
us to compute all but a finite number values of $H_{Z}$
without using the coordinates of points.
We also characterize the ACM fat point schemes
sing our description of the eventual behaviour. In fact,
n the case that $Z \subseteq \popo$ is ACM, then
the entire Hilbert function and its minimal free resolution
depend solely on knowing the eventual values of the Hilbert function.
Keywords:Hilbert function, points, fat points, CohenMacaulay, multiprojective space Categories:13D40, 13D02, 13H10, 14A15 
