ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL
ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL
复制标题
多项式的边界秩与平滑秩的差异
DOI：
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发表时间：
2013
影响因子：
0.5
通讯作者：
J. Buczýnski
中科院分区：
文献类型：
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作者：
Weronika Buczýnska;J. Buczýnski
摘要我们考虑了对Veronese变化的较高的距离变化。 。在本说明中，我们的距离各种阶段的所有点的排名。边界排名5和可平稳的排名6。我们还证明，最多四个变量的所有立方体都具有平滑等级等于边界等级。
Abstract We consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.