论文标题
多剂引理
Multi-Secant Lemma
论文作者
论文摘要
我们提出了对经典三局引理的新概括。我们的方法与以前的概括完全不同。令$ x $成为等等的投影尺寸$ d $。对于给定的$ k \ leq d + 1 $，我们对研究各种$ k $ secants的研究感兴趣。经典的三角弯曲只是考虑了$ k = 3 $而在其他地方考虑$ k = d + 2 $的情况。订单的额外订单从$ 4 $到$ d + 1 $为我们的主要结果提供服务。在本文中，我们证明，如果$ k $ secant（$ k \ leq d + 1 $）满足以下三个条件：（i）每一个$ x $中的每个点都会通过$ x $中的每个点，至少要通过一个$ k $ secant，（ii）$ k $ - secect的多样性，$ k $ -secant your quant $ k + s $ k + s $ k + s $ k + s $ - 然后，品种$ x $可以嵌入到$ p^{d+1} $中。这里介绍的新假设是我们称之为强连接性是必不可少的，因为在某个示例中，没有纳入该假设的幼稚概括失败。本文以一些关于强连通性假设的本质的猜想结束。
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.
