论文标题
Chow品种的所有割线品种对于立方体和第四纪形式都是非缺陷的
All secant varieties of the Chow variety are nondefective for cubics and quaternary forms
论文作者
论文摘要
形式的食物等级是其最小分解为线性形式产物总和的长度。对于通用形式，这对应于发现填充环境空间的最小割线品种。我们通过证明所有相关品种的非缺陷性来确定通用立方体和第四纪形式的ChOW等级。我们证明的主要新成分是[Brambilla和Ottaviani，Alexander-Hirschowitz Theorem，J。PureAppl。代数，2008年]，包括采用Terracini的引理和牛顿的落后差异公式来计算任意投射品种的割线尺寸。通过这种归纳构建，非缺陷的证明最终减少了许多基本案例。由于所涉及的空间的较大维度，这些这些是通过计算机辅助的证据来解决的。我们证明所需的最大基本案例包括计算从$ 400 $ th secant的构建的矢量空间的维度，该度量的$ 82 $ 82 $ Chow品种嵌入了$ \ Mathbb {p}^{98769} $中。
The Chow rank of a form is the length of its smallest decomposition into a sum of products of linear forms. For a generic form, this corresponds to finding the smallest secant variety of the Chow variety which fills the ambient space. We determine the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties. The main new ingredient in our proof is the generalization of a technique by [Brambilla and Ottaviani, On the Alexander--Hirschowitz theorem, J. Pure Appl. Algebra, 2008] that consists of employing Terracini's lemma and Newton's backward difference formula to compute the dimensions of secant varieties of arbitrary projective varieties. Via this inductive construction, the proof of nondefectivity ultimately reduces to proving a number of base cases. These are settled via a computer-assisted proof because of the large dimensions of the spaces involved. The largest base case required in our proof consisted of computing the dimension of a vector space constructed from the $400$th secant variety of a degree-$82$ Chow variety embedded in $\mathbb{P}^{98769}$.
