Asymptotics of degrees and ED degrees of Segre products

Giorgio Ottaviani,Luca Sodomaco,Emanuele Ventura

doi:10.1016/j.aam.2021.102242

Giorgio Ottaviani, Luca Sodomaco + Show 1 more

Open Access

https://doi.org/10.1016/j.aam.2021.102242

Copy DOI

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×Qn, where X is a projective variety and Qn⊂Pn+1 is a smooth quadric hypersurface.

Highlights

Let V R be a real vector space equipped with a distance function and let X ⊂ P (V R ⊗R C) be a complex projective variety
We start off from the very classical hyperdeterminants of the Segre variety P (Cn+1)×d establishing an asymptotic formula of its degree as the dimension n + 1 of the vector space goes to infinity
We focus on the case when V = vω1 P (V1) ⊗· · ·⊗vωd P (Vd), X = P (V1)×· · ·×P (Vd) and δ = δF

Summary

Introduction

Let V R be a real vector space equipped with a distance function and let X ⊂ P (V R ⊗R C) be a complex projective variety. We start off from the very classical (hypercubical) hyperdeterminants of the Segre variety P (Cn+1)×d establishing an asymptotic formula of its degree as the dimension n + 1 of the vector space goes to infinity. This is in contrast with the first nontrivial case of 2 × 2 × 2 tensors, where the degree of the hyperdeterminant is 4 and EDdegreeF (X) = 6 Another asymptotic result is proved for n = 1, i.e., when X is a Segre product of d projective lines, the basic space of qubits in Quantum Information Theory. The stabilization of the Frobenius ED degree of some Segre varieties is a very interesting phenomenon This is apparent from expanding Friedland-Ottaviani’s formula, which expresses the number of singular vector tuples of a general tensor in a given format.

Preliminaries

Asymptotics of degrees of some hyperdeterminants

Stabilization of the ED degree of some Segre varieties

Stabilization of the degree of the dual of a special Segre product

The ED polynomial of a Segre product of two projective varieties