Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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ON THE BETTI NUMBERS OF THREE FAT POINTS IN ℙ1 × ℙ1

  • Received : 2018.06.08
  • Accepted : 2018.10.12
  • Published : 2019.05.01
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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.

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DBSHBB_2019_v56n3_751_f0001.png 이미지

FIGURE 1. The set of 3 fat points in ℙ1 × ℙ1

DBSHBB_2019_v56n3_751_f0002.png 이미지

FIGURE 2. The set of 5 fat points in ℙ2

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