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On the Betti numbers of three fat points in $\mathbb{P}^1 \times \mathbb{P}^1$
Giuseppe Favacchio,Elena Guardo 대한수학회 2019 대한수학회지 Vol.56 No.3
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.
A NUMERICAL PROPERTY OF HILBERT FUNCTIONS AND LEX SEGMENT IDEALS
Favacchio, Giuseppe Korean Mathematical Society 2020 대한수학회지 Vol.57 No.3
We introduce the fractal expansions, sequences of integers associated to a number. We show that these sequences characterize the O-sequences and encode some information about lex segment ideals. Moreover, we introduce numerical functions called fractal functions, and we use them to solve the open problem of the classification of the Hilbert functions of any bigraded algebra.
A numerical property of Hilbert functions and lex segment ideals
Giuseppe Favacchio 대한수학회 2020 대한수학회지 Vol.57 No.3
We introduce the \textit{fractal expansions}, sequences of integers associated to a number. We show that these sequences characterize the $O$-sequences and encode some information about lex segment ideals. Moreover, we introduce numerical functions called \textit{fractal functions}, and we use them to solve the open problem of the classification of the Hilbert functions of any bigraded algebra.
ON THE BETTI NUMBERS OF THREE FAT POINTS IN ℙ<sup>1</sup> × ℙ<sup>1</sup>
Favacchio, Giuseppe,Guardo, Elena Korean Mathematical Society 2019 대한수학회지 Vol.56 No.3
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.