• 대한수학회보
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• 제58권1호
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• pp.253-267
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• 2021

We study linearly dependent subsets with prescribed cardinality s of a multiprojective space. If the set S is a circuit, there is an upper bound on the number of factors of the minimal multiprojective space containing S. B. Lovitz gave a sharp upper bound for this number. If S has higher dependency, this may be not true without strong assumptions (and we give examples and suitable assumptions). We describe the dependent subsets S with #S = 6.

• 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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• 제59권4호
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• pp.929-949
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• 2022

Given an ACM set 𝕏 of points in a multiprojective space ℙ^m×ℙⁿ over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for 𝕏. In ℙ¹×ℙ¹, the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in ℙ^m×ℙⁿ for n > 1 or m > 1. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that 𝕏 is a complete intersection of type (d₁, …, d_m, d'₁, …, d'_n) if and only if it has the Cayley-Bacharach property and the Kähler different is non-zero at a certain degree. We characterize the Cayley-Bacharach property of 𝕏 under certain assumptions.