• Title/Summary/Keyword: linear star-configurations

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THE MINIMAL FREE RESOLUTION OF THE UNION OF TWO LINEAR STAR-CONFIGURATIONS IN ℙ2

  • Shin, Yong-Su
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.683-693
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    • 2016
  • In [1], the authors proved that the finite union of linear star-configurations in $\mathbb{P}^2$ has a generic Hilbert function. In this paper, we find the minimal graded free resolution of the union of two linear star-configurations in $\mathbb{P}^2$ of type $s{\times}t$ with $\(^t_2\){\leq}s$ and $3{\leq}t$.

SOME EXAMPLES OF THE UNION OF TWO LINEAR STAR-CONFIGURATIONS IN ℙ2 HAVING GENERIC HILBERT FUNCTION

  • Shin, Yong Su
    • Journal of the Chungcheong Mathematical Society
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    • v.26 no.2
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    • pp.403-409
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    • 2013
  • In [20] and [22], the author proved that the union of two linear star-configurations in $\mathbb{P}^2$ of type $t{\times}s$ with $3{\leq}t{\leq}10$ and $t{\leq}s$ has generic Hilbert function. In this paper, we prove that the union of two linear star-configurations in $\mathbb{P}^2$ of type $t{\times}s$ with $3{\leq}t$ and $({a\atop2})-1{\leq}s$ has also generic Hilbert function.

ON THE HILBERT FUNCTION OF THE UNION OF TWO LINEAR STAR-CONFIGURATIONS IN $\mathbb{P}^2$

  • Shin, Yong Su
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.3
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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].

THE MINIMAL GRADED FREE RESOLUTION OF THE UNION OF TWO STAR CONFIGURATIONS IN 𝕡n AND THE WEAK LEFSCHETZ PROPERTY

  • Shin, Yong-Su
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.4
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    • pp.435-443
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    • 2017
  • We find a graded minimal free resolution of the union of two star configurations ${\mathbb{X}}$ and ${\mathbb{Y}}$ (not necessarily linear star configurations) in ${\mathbb{P}}^n$ of type s and t for s, $t{\geq}2$, and $n{\geq}3$. As an application, we prove that an Artinian ring $R/(I_{\mathbb{X}}+I_{\mathbb{Y}})$ of two linear star configurations ${\mathbb{X}}$ and ${\mathbb{Y}}$ in ${\mathbb{P}}^3$ of type s and t has the weak Lefschetz property for $s{\geq}{\lfloor}\frac{1}{2}(^t_2){\rfloor}$ and $t{\geq}2$.

SOME APPLICATIONS OF THE UNION OF STAR-CONFIGURATIONS IN ℙn

  • Shin, Yong Su
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.4
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    • pp.807-824
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    • 2011
  • It has been proved that if $\mathbb{X}^{(s,s)}$ is the union of two linear star-configurations in $\mathbb{P}^2$ of type $s{\times}s$, then $(I_{\mathbb{X}^{(s,s)}})_s{\neq}\{0\}$ for s = 3, 4, 5, and $(I_{\mathbb{X}^{(s,s)}})_s=\{0\}$ for $s{\geq}6$. We extend $\mathbb{P}^2$ to $\mathbb{P}^n$ and show that if $\mathbb{X}^{(s,s)}$ is the union of two linear star-configurations in $\mathbb{P}^n$, then $(I_{\mathbb{X}^{(s,s)}})_s=\{0\}$ for $n{\geq}3$ and $s{\geq}3$. Using this generalization, we also prove that the secant variety $Sec_1(Split_s(\mathbb{P}^n))$ has the expected dimension 2ns + 1 for $n{\geq}3$ and $s{\geq}3$.

THE ARTINIAN POINT STAR CONFIGURATION QUOTIENT AND THE STRONG LEFSCHETZ PROPERTY

  • Kim, Young-Rock;Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.56 no.3
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    • pp.645-667
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    • 2019
  • It has been little known when an Artinian point quotient has the strong Lefschetz property. In this paper, we find the Artinian point star configuration quotient having the strong Lefschetz property. We prove that if ${\mathbb{X}}$ is a star configuration in ${\mathbb{P}}^2$ of type s defined by forms (a-quadratic forms and (s - a)-linear forms) and ${\mathbb{Y}}$ is a star configuration in ${\mathbb{P}}^2$ of type t defined by forms (b-quadratic forms and (t - b)-linear forms) for $b=deg({\mathbb{X}})$ or $deg({\mathbb{X}})-1$, then the Artinian ring $R/(I{\mathbb_{X}}+I{\mathbb_{Y}})$ has the strong Lefschetz property. We also show that if ${\mathbb{X}}$ is a set of (n+ 1)-general points in ${\mathbb{P}}^n$, then the Artinian quotient A of a coordinate ring of ${\mathbb{X}}$ has the strong Lefschetz property.

THE MINIMAL FREE RESOLUTION OF A STAR-CONFIGURATION IN ?n AND THE WEAK LEFSCHETZ PROPERTY

  • Ahn, Jea-Man;Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.49 no.2
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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.

A GRADED MINIMAL FREE RESOLUTION OF THE 2ND ORDER SYMBOLIC POWER OF THE IDEAL OF A STAR CONFIGURATION IN ℙn

  • Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.56 no.1
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    • pp.169-181
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    • 2019
  • In [9], Geramita, Harbourne, and Migliore find a graded minimal free resolution of the 2nd order symbolic power of the ideal of a linear star configuration in ${\mathbb{P}}^n$ n of any codimension r. In [8], Geramita, Galetto, Shin, and Van Tuyl extend the result on a general star configuration in ${\mathbb{P}}^n$ but for codimension 2. In this paper, we find a graded minimal free resolution of the 2nd order symbolic power of the ideal of a general star configuration in ${\mathbb{P}}^n$ of any codimension r using a matroid configuration in [10]. This generalizes both the result on a linear star configuration in ${\mathbb{P}}^n$ of codimension r in [9] and the result on a general star configuration in ${\mathbb{P}}^n$ of codimension 2 in [8].

AN ARTINIAN POINT-CONFIGURATION QUOTIENT AND THE STRONG LEFSCHETZ PROPERTY

  • Kim, Young Rock;Shin, Yong-Su
    • Journal of the Korean Mathematical Society
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    • v.55 no.4
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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 $\mathbb{p}^2$.