Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE ARTINIAN POINT STAR CONFIGURATION QUOTIENT AND THE STRONG LEFSCHETZ PROPERTY

  • Kim, Young-Rock (Major in Mathematics Education Graduate School of Education Hankuk University of Foreign Studies) ;
  • Shin, Yong-Su (Department of Mathematics Sungshin Women's University)
  • Received : 2018.04.18
  • Accepted : 2019.01.08
  • Published : 2019.05.01
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Abstract

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.

Keywords

DBSHBB_2019_v56n3_645_f0001.png 이미지

FIGURE 1. The union of two point star configurations in ℙ2 of types 3 and 5

TABLE 1. A 𝕜-configuration in ℙ2 of type (1, 2, 3, . . . , t + b -2, t+ b- 1, t+ b)

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TABLE 2. A set 𝕏 ∪ 𝕐 = 𝕌 ∪ 𝕍

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TABLE 3. A 𝕜-configuration ∪ in ℙ2 of type (t+b-2u, . . . , t+b- 2, t+ b- 1)

DBSHBB_2019_v56n3_645_t0003.png 이미지

TABLE 4. A point star configuration 𝕍 in ℙ2 of type (t - u)defined by (b- u)-quadratic forms and (t- b)-linear forms

DBSHBB_2019_v56n3_645_t0004.png 이미지

TABLE 5. A 𝕜-configuration in ℙ2 of type (1, 2, 3, ..., $\underline{t-u+2b-1}$, $\underline{t-u+2b+1}$, ..., t+b)

DBSHBB_2019_v56n3_645_t0005.png 이미지

TABLE 6. A 𝕜-configuration in ℙ2 of type (1, 2, 3, ..., $\underline{t+b-2l-1}$, $\underline{t+b-2l+1}$, ..., t+b)

DBSHBB_2019_v56n3_645_t0006.png 이미지

TABLE 7. A 𝕜-configuration in ℙ2 of type (1, 2, 3, ..., $\underline{t+b-2l-2}$, $\underline{t+b-2l}$, ... , t+b)

DBSHBB_2019_v56n3_645_t0007.png 이미지

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