• 한국수학교육학회지시리즈B:순수및응용수학
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• 제25권1호
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• pp.31-38
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• 2018

In [4], the authors show that if ${\mathbb{X}}$ is a ${\mathbb{k}}-configuration$ in ${\mathbb{P}}^2$ of type ($d_1$, ${\ldots}$, $d_s$) with $d_s$ > $s{\geq}2$, then ${\Delta}H_{m{\mathbb{X}}}(md_s-1)$ is the number of lines containing exactly $d_s-points$ of ${\mathbb{X}}$ for $m{\geq}2$. They also show that if ${\mathbb{X}}$ is a ${\mathbb{k}}-configuration$ in ${\mathbb{P}}^2$ of type (1, 2, ${\ldots}$, s) with $s{\geq}2$, then ${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)$ is the number of lines containing exactly s-points in ${\mathbb{X}}$ for $m{\geq}s+1$. In this paper, we explore a standard ${\mathbb{k}}-configuration$ in ${\mathbb{P}}^2$ and find that if ${\mathbb{X}}$ is a standard ${\mathbb{k}}-configuration$ in ${\mathbb{P}}^2$ of type (1, 2, ${\ldots}$, s) with $s{\geq}2$, then ${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)=3$, which is the number of lines containing exactly s-points in ${\mathbb{X}}$ for $m{\geq}2$ instead of $m{\geq}s+1$.

• 대한수학회지
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• 제55권4호
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• pp.763-783
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• 2018

In this paper, we study an Artinian point-configuration quotient having the SLP. We show that an Artinian quotient of points in $\mathbb{p}^n$ has the SLP when the union of two sets of points has a specific Hilbert function. As an application, we prove that an Artinian linear star configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ and $\mathbb{Y}$ are linear starconfigurations in $\mathbb{p}^2$ of type s and t for $s{\geq}(^t_2)-1$ and $t{\geq}3$. We also show that an Artinian $\mathbb{k}$-configuration quotient $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ has the SLP if $\mathbb{X}$ is a $\mathbb{k}$-configuration of type (1, 2) or (1, 2, 3) in $\mathbb{p}^2$, and $\mathbb{X}{\cup}\mathbb{Y}$ is a basic configuration in $\mathbb{p}^2$.

• 대한수학회지
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• 제49권2호
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• pp.405-417
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• 2012

We find the Hilbert function and the minimal free resolution of a star-configuration in $\mathbb{P}^n$. The conditions are provided under which the Hilbert function of a star-configuration in $\mathbb{P}^2$ is generic or non-generic We also prove that if $\mathbb{X}$ and $\mathbb{Y}$ are linear star-configurations in $\mathbb{P}^2$ of types t and s, respectively, with $s{\geq}t{\geq}3$, then the Artinian k-algebra $R/(I_{\mathbb{X}}+I_{\mathbb{Y})$ has the weak Lefschetz property.

• Journal of applied mathematics & informatics
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• 제2권1호
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• pp.59-83
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• 1995

In this paper, we proved the set of points which are the vertices of the n-gon in $mathbb{P}^2(n\geq3$)$ has the Uniform Position Property and what the graded free resolutions of the ideals of k-configurations in $mathbb{P}^3$ are.

• 충청수학회지
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• 제27권3호
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• pp.413-418
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• 2014

It has been proved that the union of two linear star-configurations in $\mathbb{P}^2$ of type s and t for either $3{\leq}t{\leq}10$ or $\(\frac{t}{2}\)-1{\leq}s$ with $3{\leq}t$ has maximal Hilbert function. We extend the condition to $\[\frac{1}{2}\(\frac{t}{2}\)\]{\leq}s$, so that it is true for either $3{\leq}t{\leq}10$ or $\[\frac{1}{2}\(\frac{t}{2}\)\]{\leq}s$ with $3{\leq}t$, which extends the result of [6].

• 충청수학회지
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• 제28권1호
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• pp.119-125
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• 2015

We find a necessary and sufficient condition for which a point star-configuration in $\mathbb{P}^n$ has generic Hilbert function. More precisely, a point star-configuration in $\mathbb{P}^n$ defined by general forms of degrees $d_1,{\ldots},d_s$ with $3{\leq}n{\leq}s$ has generic Hilbert function if and only if $d_1={\cdots}=d_{s-1}=1$ and $d_s=1,2$. Otherwise, the Hilbert function of a point star-configuration in $\mathbb{P}^n$ is NEVER generic.

• Journal of applied mathematics & informatics
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• 제4권1호
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• pp.249-261
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• 1997

We find a necessary and sufficient condition for a $\kappa$-confi-guration $\mathbb{X}$ in $\mathbb{P}^2$ to be in generic position. We obtain the number and degrees of minimal generators of some Gorenstein ideals of codimension 3 and so obtain their minimal free resolution s of these ideals.

• 충청수학회지
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• 제25권3호
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• pp.553-562
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• 2012

It has been proved that the union of two linear star-configurations in $\mathbb{P}^2$ of type $t{\times}s$ for $3{\leq}t{\leq}9$ and $3{\leq}t{\leq}s$ has generic Hilbert function. We extend the condition to $t$ = 10, so that it is true for $3{\leq}t{\leq}10$, which generalizes the result of [7].