On the Betti numbers of three fat points in $\mathbb{P}^1 \times \mathbb{P}^1$
J. Korean Math. Soc. 2019 Vol. 56, No. 3, 751-766
https://doi.org/10.4134/JKMS.j180385
Published online May 1, 2019
Giuseppe Favacchio, Elena Guardo
Dipartimento di Matematica e Informatica; Dipartimento di Matematica e Informatica
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.
Keywords : multiprojective spaces, Hilbert functions, fat points
MSC numbers : 13F20, 13A15, 13D40, 14M05
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