Asymptotics of degrees and ED degrees of Segre products

TL;DR

This paper investigates the asymptotic behavior of algebraic and Euclidean Distance degrees of Segre products and their duals, providing geometric insights and stabilization results for these invariants.

Contribution

It offers a new geometric perspective on the stabilization of ED degrees and degrees of dual varieties of Segre products, including hyperdeterminants and Segre-quadric products.

Findings

Asymptotic formulas for degrees of hyperdeterminants.

Geometric explanation for ED degree stabilization.

Proof of degree stabilization for Segre-quadric 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.