대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제54권4호
  • /
  • Pages.1323-1330
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  • 2017
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON MULTISECANT PLANES OF LOCALLY NON-COHEN-MACAULAY SURFACES

  • Lee, Wanseok (Department of Applied Mathematics Pukyong National University) ;
  • Park, Euisung (Department of Mathematics Korea University)
  • 투고 : 2016.07.06
  • 심사 : 2017.02.23
  • 발행 : 2017.07.31
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⟨ 이전 논문 다음 논문 ⟩

초록

For a nondegenerate projective irreducible variety $X{\subset}{\mathbb{P}}^r$, it is a natural problem to find an upper bound for the value of $${\ell}_{\beta}(X)=max\{length(X{\cap}L){\mid}L={\mathbb{P}}^{\beta}{\subset}{\mathbb{P}}^r,\;{\dim}(X{\cap}L)=0\}$$ for each $1{\leq}{\beta}{\leq}e$. When X is locally Cohen-Macaulay, A. Noma in [10] proves that ${\ell}_{\beta}(X)$ is at most $d-e+{\beta}$ where d and e are respectively the degree and the codimension of X. In this paper, we construct some surfaces $S{\subset}{\mathbb{P}}^5$ of degree $d{\in}\{7,{\ldots},12\}$ which satisfies the inequality $${\ell}_2(S){\geq}d-3+{\lfloor}{\frac{d}{2}}{\rfloor}$$. This shows that Noma's bound is no more valid for locally non-Cohen-Macaulay varieties.

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참고문헌

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