论文标题
投影品品种的速度指数
Secant indices of projective varieties
论文作者
论文摘要
对于每个子变量$ x $，投影$ n $ -n $ - codimension $ m $的空间，我们将一个长度$ m + 1 $从$ 1 $到$ x $的整数序列，记录有限的最大基础性，减少了$ x $的交叉点与线性subvarieties。我们将其称为$ x $的速度指数的顺序。已经独立研究了类似的数字，目的是将亚属体分类为极端的距离空间。我们在本说明中的重点是对割线指数共同满足的组合特性的研究。我们表明，对于非平滑平滑的亚变化，这些序列严格增加，开发了一种计算速度指数的项下限的方法，并计算了Veronese和Segre品种的这些下限。就Veronese品种而言，Eisenbud-Green-Harris猜想的真相将暗示我们发现的下限实际上等于割线指数。在此过程中，我们陈述了几个相关问题和其他猜想，据我们所知，这是开放的。
To each subvariety $X$ in projective $n$-space of codimension $m$ we associate an integer sequence of length $m + 1$ from $1$ to the degree of $X$ recording the maximal cardinalities of finite, reduced intersections of $X$ with linear subvarieties. We call this the sequence of secant indices of $X$. Similar numbers have been studied independently with the aim of classifying subvarieties with extremal secant spaces. Our focus in this note is the study of the combinatorial properties that the secant indices satisfy collectively. We show these sequences are strictly increasing for nondegenerate smooth subvarieties, develop a method to compute term-wise lower bounds for the secant indices, and compute these lower bounds for Veronese and Segre varieties. In the case of Veronese varieties, the truth of the Eisenbud-Green-Harris conjecture would imply the lower bounds we find are in fact equal to the secant indices. Along the way we state several relevant questions and additional conjectures which to our knowledge are open.
