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

Giuseppe Favacchio; Elena Guardo

doi:10.4134/JKMS.j180385

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2019; 56(3): 751-766

Online first article February 1, 2019 Printed May 1, 2019

https://doi.org/10.4134/JKMS.j180385

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

Giuseppe Favacchio, Elena Guardo

Dipartimento di Matematica e Informatica; Dipartimento di Matematica e Informatica

Abstract

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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