On the Hilbert function on $\mathbb P^1\times\mathbb P^1$

Let $Q = \mathbb P^1 x \mathbb P^1$ and let $X\subset Q$ be a 0-dimensional scheme. This paper is a first step towards the characterization of Hilbert functions of 0- dimensional schemes in $Q$. In particular we show how, under some conditions on $X$, its Hilbert function changes when we add points to $X$ lying on a $(1,0)$ or $(0, 1)$-line. As a particular case we show also that if $X$ is ACM this result holds without any additional hypothesis.