$\bold P^5$中曲线的一个特征定理
A speciality theorem for curves in $\bold P^5$
摘要：
设$C\subset \bold P^r$是一整射影曲线。定义$C$的特殊性指标$e(C)$为最大整数$t$使得$h^0(C,ω_C(-t))>0$，其中$ω_C$表示$C$的对偶层。本文推广了Hüssen关于空间曲线特殊性的一个经典结果，证明了：如果$C\subset \bold P^5$是整数次$d$曲线，不包含在任何$< s$次曲面、任何三次$<t$次曲面和任何四次$<u$次曲面中，且如果$d>>s>>t>>u\geq 1$，然后$ e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. $此外等式成立当且仅当$C$是次数为$u$，${\frac{t}{u}}$，${\frac{s}{t}}$和${\frac{d}{s}}$的超曲面的完全交。在一般情形$C\subset \bold P^r$，$r\geq 3$中，我们也给出了部分结果.
英文摘要：
Let $C\subset \bold P^r$ be an integral projective curve. One defines the speciality index $e(C)$ of $C$ as the maximal integer $t$ such that $h^0(C,ω_C(-t))>0$, where $ω_C$ denotes the dualizing sheaf of $C$. Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if $C\subset \bold P^5$ is an integral degree $d$ curve not contained in any surface of degree $< s$, in any threefold of degree $<t$, and in any fourfold of degree $<u$, and if $d>>s>>t>>u\geq 1$, then $ e(C)\leq {\frac{d}{s}}+{\frac{s}{t}}+{\frac{t}{u}}+u-6. $ Moreover equality holds if and only if $C$ is a complete intersection of hypersurfaces of degrees $u$, ${\frac{t}{u}}$, ${\frac{s}{t}}$ and ${\frac{d}{s}}$. We give also some partial results in the general case $C\subset \bold P^r$, $r\geq 3$.
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