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Asymptotics of degrees and ED degrees of Segre products
Advances in Applied Mathematics ( IF 1.0 ) Pub Date : 2021-06-17 , DOI: 10.1016/j.aam.2021.102242
Giorgio Ottaviani , Luca Sodomaco , Emanuele Ventura

Two fundamental invariants attached to a projective variety are its classical algebraic degree and its Euclidean Distance degree (ED degree). In this paper, we study the asymptotic behavior of these two degrees of some Segre products and their dual varieties. We analyze the asymptotics of degrees of (hypercubical) hyperdeterminants, the dual hypersurfaces to Segre varieties.

We offer an alternative viewpoint on the stabilization of the ED degree of some Segre varieties. Although this phenomenon was incidentally known from Friedland-Ottaviani's formula expressing the number of singular vector tuples of a general tensor, our approach provides a geometric explanation.

Finally, we establish the stabilization of the degree of the dual variety of a Segre product X×Qn, where X is a projective variety and QnPn+1 is a smooth quadric hypersurface.



中文翻译:

Segre 乘积的度数和 ED 度数的渐近性

附加到射影变数的两个基本不变量是其经典代数度和欧几里得距离度(ED 度)。在本文中,我们研究了一些 Segre 产品及其对偶品种的这两个度的渐近行为。我们分析了(超立方)超行列式的度数的渐近性,即 Segre 变体的双重超曲面。

我们对某些 Segre 品种的 ED 度的稳定性提出了另一种观点。尽管从 Friedland-Ottaviani 公式中偶然知道了这种现象,该公式表示一般张量的奇异向量元组的数量,但我们的方法提供了几何解释。

最后,我们建立了一个 Segre 产品的二元变异度的稳定性 X×n,其中X是一个射影变种,并且nn+1 是光滑的二次超曲面。

更新日期:2021-06-17
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