On the Betti numbers of three fat points in $\mathbb{P}^1 \times \mathbb{P}^1$

다국어 초록 (Multilingual Abstract)

In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set $Z$ of three fat points whose support is an almost complete intersection (ACI) in $\mathbb{P}^1 \times \mathbb{P}^1.$ A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in $\mathbb P^2$ and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.

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