On the Betti numbers of three fat points in $\mathbb{P}^1\times \mathbb{P}^1$
J. Korean Math. Soc.
Published online 2019 Feb 01
Giuseppe Favacchio, and Elena Guardo
Università di Catania
Abstract : In these notes we introduce a numerical function which allow us to describe explicitely (and not recursively) 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 not recursively 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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