Asymptotics of degrees and ED degrees of Segre products

A Segre product of projective spaces X describes all decomposable tensors with a prescribed format. When the dual variety of X is a hypersurface, its defining polynomial is the classical hyperdeterminant. In the first part of this talk, we discuss the asymptotic behavior of the degree of the hyperdeterminant for tensors of hypercubical format.

Another fundamental invariant attached to the Segre product X is its Euclidean Distance Degree (ED degree), which measures the complexity of minimizing the distance from X. Friedland and Ottaviani derived a beautiful formula for the ED degree of X and observed that this invariant stabilizes as soon as the dimension of one of the factors is large enough. In the second part of this talk, we give a more geometric explanation of this fact.

In the last part, we consider the Segre product Z between a projective variety Y and a smooth quadric hypersurface Q. We discuss the stabilization of the degree of the dual variety of Z as soon as the dimension of the quadric is large enough. This is joint work with Giorgio Ottaviani and Emanuele Ventura.