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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.
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참고문헌 (11)
참고문헌 신청
M.-A. Bertin / 2002 / On the regularity of varieties having an extremal secant line / J. Reine Angew. Math 545:167~181
M. Brodmann / 2017 / Projective varieties of maximal sec-tional regularity / J. Pure Appl. Algebra 221(1):98~118
M. Brodmann / 2017 / On surfaces of maximal sectional regularity / Taiwanese J. Math 21(3):549~567
M. Brodmann / 2012 / Projective surfaces of degree r + 1 in projective r-space and almost non-singular projections / J. Pure Appl. Algebra 2016(10):2241~2255
M. Decker / Singular 3−1−2 – A computer algebra system for polynomial computations
M. Brodmann / 2017 / Projective varieties of maximal sec-tional regularity / J. Pure Appl. Algebra 221(1):98~118
M. Brodmann / 2017 / On surfaces of maximal sectional regularity / Taiwanese J. Math 21(3):549~567
M. Brodmann / 2012 / Projective surfaces of degree r + 1 in projective r-space and almost non-singular projections / J. Pure Appl. Algebra 2016(10):2241~2255
M. Decker / Singular 3−1−2 – A computer algebra system for polynomial computations
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