Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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SOME APPLICATIONS OF THE UNION OF STAR-CONFIGURATIONS IN ℙn

  • Shin, Yong Su (Department of Mathematics Sungshin Women's University)
  • Received : 2011.08.27
  • Accepted : 2011.11.18
  • Published : 2011.12.30
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Abstract

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 $s{\geq}3$.

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Acknowledgement

Supported by : Sungshin Women's University

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