完全可分解形式簇的无亏割线簇
Nondefective secant varieties of varieties of completely decomposable forms
摘要：
一个变化的华林的问题从经典数论的问题，"什么是最小的数字$s$，使任何通用齐次多项式的次数$d$在$n+1$变量可以写为总和的最多$s$产品的线性形式？这个问题可以通过确定最小的$s$来几何地回答，使得各种完全可分解形式的$s$ \nth正割种类填充周围空间。如果这个割线品种具有预期的尺寸，它被称为无缺陷的，和$s=\left\lceil\binom{n+d}{d}/(dn+1)\right\rceil$。证明了割线簇总是无亏的，除非是$d=2$和$2\leq s\leq\frac{n}{2}$ .我们证明了这个猜想的几个特殊情况。特别地，我们定义函数$s_1$和$s_2$，使得当$n\geq 3$和$s\leq s_1(d)$或当$n=3$和$s\geq s_2(d)$时正割簇是无缺陷的，以及函数$c$，使得当$d\geq n\geq 4$和$s\leq 2^{n-3}c(n,d)$时正割簇是无缺陷的。我们进一步证明了当$s\leq 30$时，割线簇是无缺陷的，除非$d=2$和$2\leq s\leq\frac{n}{2}$。
英文摘要：
A variation of Waring's problem from classical number theory is the question, ``What is the smallest number $s$ such that any generic homogeneous polynomial of degree $d$ in $n+1$ variables may be written as the sum of at most $s$ products of linear forms?'' This question may be answered geometrically by determining the smallest $s$ such that the $s$\nth secant variety of the variety of completely decomposable forms fills the ambient space. If this secant variety has the expected dimension, it is called nondefective, and $s=\left\lceil\binom{n+d}{d}/(dn+1)\right\rceil$. It is conjectured that the secant variety is always nondefective unless $d=2$ and $2\leq s\leq\frac{n}{2}$. We prove several special cases of this conjecture. In particular, we define functions $s_1$ and $s_2$ such that the secant variety is nondefective when $n\geq 3$ and $s\leq s_1(d)$ or when $n=3$ and $s\geq s_2(d)$ and a function $c$ such that the secant variety is nondefective when $d\geq n\geq 4$ and $s\leq 2^{n-3}c(n,d)$. We further show that the secant variety is nondefective when $s\leq 30$ unless $d=2$ and $2\leq s\leq\frac{n}{2}$.
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