On multisecant planes of locally non-Cohen-Macaulay surfaces
Bull. Korean Math. Soc. 2017 Vol. 54, No. 4, 1323-1330
Published online July 31, 2017
Wanseok Lee and Euisung Park
Pukyong National University, Korea University
Abstract : For a nondegenerate projective irreducible variety $X \subset \P^r$, it is a natural problem to find an upper bound for the value of \begin{equation*} \ell_{\beta} (X) = {\rm max} \{ \rm{length}(X \cap L)~|~ L= \P^{\beta} \subset \P^r ,~ {\rm dim}~(X \cap L) = 0 \} \end{equation*} for each $1 \leq \beta \leq e$. When $X$ is locally Cohen-Macaulay, A. Noma in \cite{N} 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 $S \subset \P^5$ of degree $d \in \{7,\ldots ,12 \}$ which satisfies the inequality \begin{equation*} \ell_2 (S) \geq d-3+\lfloor \frac{d}{2} \rfloor. \end{equation*} This shows that Noma's bound is no more valid for locally non-Cohen-Macaulay varieties.
Keywords : multisecant space, locally Cohen-Macaulayness, rational surface
MSC numbers : Primary 14C17, 14M20, 14Q10
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