• 충청수학회지
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• 제27권1호
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• pp.39-46
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• 2014

We completely classify the dimension of secant varieties $Sec_1(\mathbb{X}_{{\lambda},2})$ to the variety of reducible forms in $\mathbf{k}[x_0,x_1,x_2]$ when ${\lambda}=(1,{\cdots},1,3,{\cdots},3$), and also show that they are all non-defective.

• 대한수학회논문집
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• 제38권1호
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• pp.47-53
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• 2023

The Terracini t-locus of an embedded variety X ⊂ ℙ^r is the set of all cardinality t subsets of the smooth part of X at which a certain differential drops rank, i.e., the union of the associated double points is linearly dependent. We give an easy to check criterion to exclude some sets from the Terracini loci. This criterion applies to tensors and partially symmetric tensors. We discuss the non-existence of codimension 1 Terracini t-loci when t is the generic X-rank.

• 대한수학회보
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• 제57권4호
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• pp.865-872
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• 2020

Let X ⊂ ℙ^r be an integral and non-degenerate variety. Set n := dim(X). Let 𝜌(X)" be the maximal integer such that every zero-dimensional scheme Z ⊂ X smoothable in X is linearly independent. We prove that X is linearly normal if 𝜌(X)" ≥ 2⌈(r + 2)/2⌉ and that 𝜌(X)" < 2⌈(r + 1)/(n + 1)⌉, unless either n = r or X is a rational normal curve.

• 충청수학회지
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• 제24권4호
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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 $s{\geq}3$.