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arXiv math/0111039math.AG
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algebraic varieties上的线路

Lines on algebraic varieties

J. M. Landsberg

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3 pages
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摘要：

如果存在有限条包含在$X$中的线通过每个一般点，则簇$X$被线覆盖。我证明了两个定理。定理1：设$X^n\subset P^M$是一个被线覆盖的簇。则至多有$n!$线通过$X$的一般点。定理2：设$X^n\subsetP^{n+1}$是超曲面，$x\in X$是一般点。如果在$x$处与$X$有联系以订购$k$的行的集合具有大于预期的维度，则有联系以订购$k$的行实际上包含在$X$中。

英文摘要：

A variety $X$ is covered by lines if there exist a finite number of lines contained in $X$ passing through each general point. I prove two theorems. Theorem 1:Let $X^n\subset P^M$ be a variety covered by lines. Then there are at most $n!$ lines passing through a general point of $X$. Theorem 2:Let $X^n\subsetP^{n+1}$ be a hypersurface and let $x\in X$ be a general point. If the set of lines having contact to order $k$ with $X$ at $x$ is of dimension greater than expected, then the lines having contact to order $k$ are actually contained in $X$.

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