• 제목/요약/키워드: generic Hilbert function

검색결과 9건 처리시간 0.02초

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

  • Shin, Yong Su
    • 충청수학회지
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    • 제26권2호
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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.

A POINT STAR-CONFIGURATION IN ℙn HAVING GENERIC HILBERT FUNCTION

  • Shin, Yong-Su
    • 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.

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

  • Shin, Yong-Su
    • 대한수학회논문집
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    • 제31권4호
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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 $3{\leq}t$.

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

  • Shin, Yong Su
    • 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].

SOME GEOMETRIC PROPERTIES OF GOTZMANN COEFFICIENTS

  • Jeaman Ahn
    • 충청수학회지
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    • 제37권2호
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    • pp.57-66
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    • 2024
  • In this paper, we study how the Hilbert polynomial, associated with a reduced closed subscheme X of codimension 2 in ℙN, reveals geometric information about X. Although it is known that the Hilbert polynomial can tell us about the scheme's degree and arithmetic genus, we find additional geometric information it can provide for smooth varieties of codimension 2. To do this, we introduce the concept of Gotzmann coefficients, which helps to extract more information from the Hilbert polynomial. These coefficients are based on the binomial expansion of values of the Hilbert function. Our method involves combining techniques from initial ideals and partial elimination ideals in a novel way. We show how these coefficients can determine the degree of certain geometric features, such as the singular locus appearing in a generic projection, for smooth varieties of codimension 2.

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

  • Ahn, Jea-Man;Shin, Yong-Su
    • 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.

NON-EXISTENCE OF SOME ARTINIAN LEVEL O-SEQUENCES OF CODIMENSION 3

  • Shin, Dong-Soo
    • Journal of applied mathematics & informatics
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    • 제23권1_2호
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    • pp.517-523
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    • 2007
  • Let R/I be an Artinian algebra of codimension 3 with Hilbert function H such that $h_{d-1}>h_d=h_{d+1}$. Ahn and Shin showed that A cannot be level if ${\beta}_{1,d+2}(Gin(I))={\beta}_{2,d+2}(Gin(I))$ where Gin(I) is a generic initial ideal of I. We prove that some certain graded Artinian algebra R/I cannot be level if either ${\beta}_{1,d}(I^{lex})={\beta}_{2,d}(I^{lex})+1\;or\;{\beta}_{1,d+1}(I^{lex})={\beta}_{2,d+1}(I^{lex})\;where\;I^{lex}$ is a lex-segment ideal associated to I.

$\kappa$-CONFIGURATIONS IN $\mathbb{P}^2$ AND GORENSTEIN IDEALS OF CODIMENSION 3

  • Shin, Yong-Su
    • 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.