Asymptotics of degrees and ED degrees of Segre products

Abstract

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 $X\times Q_n$, where $X$ is a projective variety and $Q_n\subset {\mathbb{P}}^{n+1}$ is a smooth quadric hypersurface.

arXiv comment24 pages

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Cited by in corpus (5)

• On tensors that are determined by their singular tuples
• Degrees of Kalman varieties of tensors
• A vector bundle approach to Nash equilibria
• On the minimal algebraic complexity of the rank-one approximation problem for general inner products
• The span of singular tuples of a tensor beyond the boundary format