多割线引理
Multi-Secant Lemma
摘要：
我们提出了一个新的推广经典的三割引理。我们的方法与以前的概括有很大的不同。设$X$是维数为$d$的等维射影簇.对于给定的$k \leq d + 1$，我们感兴趣的是研究$k$-割线的多样性。经典的三割线引理只考虑$k = 3$的情况，而在其他地方则考虑$k = d + 2$的情况。从$4$到$d + 1$的序割线为我们的主要结果提供了服务。本文证明了如果$k$-割线簇（$k \leq d + 1$）满足以下三个条件：（i）通过$X$中的每个点，至少通过一个$k$-割线，（ii）$k$-割线的簇满足我们在续集中定义的强连通性，（iii）每个$k$-割线也是一个（$k+1$）-正割，则变种$X$可以嵌入到$P^{d+1}$中。这里引入的我们称为强连通性的新假设是必不可少的，因为不包含这个假设的天真推广失败了，正如我们在一些例子中所展示的那样。文章最后对强连通性假设的实质作了一些说明。
英文摘要：
We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations. Let $X$ be an equidimensional projective variety of dimension $d$. For a given $k \leq d + 1$, we are interested in the study of the variety of $k$-secants. The classical trisecant lemma just considers the case where $k = 3$ while elsewhere the case $k = d + 2$ is considered. Secants of order from $4$ to $d + 1$ provide service for our main result. In this paper, we prove that if the variety of $k$-secants ($k \leq d + 1$) satisfies the three following conditions: (i) trough every point in $X$, passes at least one $k$-secant, (ii) the variety of $k$-secant satisfies a strong connectivity property that we defined in the sequel, (iii) every $k$-secant is also a ($k+1$)-secant, then the variety $X$ can be embedded into $P^{d+1}$. The new assumption, introduced here, that we called strong connectivity is essential because a naive generalization that does not incorporate this assumption fails as we show in some example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption.
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