摘要翻译：
经典的三等分引理指出，如果$x$是$pp^3$的积分曲线，则三等分簇的维数为1，除非该曲线是平面的且度数至少为3，在这种情况下，三等分簇的维数为2。在本文中，我们的目的是首先给出这个结果的另一个推导，然后引入非等维变式的一个推广。为了清楚起见，我们将把我们的第一个问题重新表述如下。设$z$是维数$n$的等维变体（可能是奇异的和/或可约的），而不是线性空间，嵌入到$\pp^r$,$r\geq n+1$中。除非$Z$包含在$(n+1)-$维线性空间中且度至少为3,否则$Z$的三等分线的变种，例如$V_{1,3}(Z)$的维数严格小于$2n$,在这种情况下$\dim(V_{1,3}(Z))=2n$。然后我们询问更一般的情况，其中$z$不要求是等维的。在这种情况下，设$z$是维数$n$的一个可能奇异的变体，它可能既不是不可约的，也不是等维的，嵌入到$\pp^r$中，其中$r\geq n+1$和$y$是维数$k\geq1$的适当子变体。现在考虑$S$是$\{l\in\g(1，r)\vtl\存在p\in Y,q_1,q_2\in Z\反斜杠Y,q_1,q_2,p\in l\}$闭包的最大维数的一个分量。我们证明$S$的维数严格小于$n+k$，除非$S$中的并线维数为$n+1$，在这种情况下$dim(S)=n+k$。在后一种情况下，如果空间的维数严格大于$n+1$，则$s$中的并线不能覆盖整个空间。这是我们工作的主要成果。我们还介绍了一些例子，说明我们的界限是严格的。
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英文标题：
《Trisecant Lemma for Non Equidimensional Varieties》
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作者：
J.Y. Kaminski, A. Kanel-Belov and M. Teicher
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最新提交年份：
2007
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  The classic trisecant lemma states that if $X$ is an integral curve of $\PP^3$ then the variety of trisecants has dimension one, unless the curve is planar and has degree at least 3, in which case the variety of trisecants has dimension 2. In this paper, our purpose is first to present another derivation of this result and then to introduce a generalization to non-equidimensional varities. For the sake of clarity, we shall reformulate our first problem as follows. Let $Z$ be an equidimensional variety (maybe singular and/or reducible) of dimension $n$, other than a linear space, embedded into $\PP^r$, $r \geq n+1$. The variety of trisecant lines of $Z$, say $V_{1,3}(Z)$, has dimension strictly less than $2n$, unless $Z$ is included in a $(n+1)-$dimensional linear space and has degree at least 3, in which case $\dim(V_{1,3}(Z)) = 2n$. Then we inquire the more general case, where $Z$ is not required to be equidimensional. In that case, let $Z$ be a possibly singular variety of dimension $n$, that may be neither irreducible nor equidimensional, embedded into $\PP^r$, where $r \geq n+1$, and $Y$ a proper subvariety of dimension $k \geq 1$. Consider now $S$ being a component of maximal dimension of the closure of $\{l \in \G(1,r) \vtl \exists p \in Y, q_1, q_2 \in Z \backslash Y, q_1,q_2,p \in l\}$. We show that $S$ has dimension strictly less than $n+k$, unless the union of lines in $S$ has dimension $n+1$, in which case $dim(S) = n+k$. In the latter case, if the dimension of the space is stricly greater then $n+1$, the union of lines in $S$ cannot cover the whole space. This is the main result of our work. We also introduce some examples showing than our bound is strict.
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PDF链接：
https://arxiv.org/pdf/0712.3878