Entry loci and ranks

TL;DR

This paper investigates the properties of entry loci of varieties, especially their irreducibility and rank characteristics, with new insights into low-degree surfaces and a class of varieties with specific rank properties.

Contribution

It introduces a class of varieties where the generic rank matches the rank of the general entry locus and shows all smooth irreducible projective varieties can be embedded to have this property.

Findings

Entry loci of low degree surfaces analyzed using Segre points.

A new class of varieties with matching generic and entry locus ranks is introduced.

Any smooth irreducible projective variety can be embedded to exhibit this property.

Abstract

We study entry loci of varieties and their irreducibility from the perspective of $X$-ranks with respect to a projective variety $X$. These loci are the closures of the points that appear in an $X$-rank decomposition of a general point in the ambient space. We look at entry loci of low degree normal surfaces in $\mathbb P^4$ using Segre points of curves; the smooth case was classically studied by Franchetta. We introduce a class of varieties whose generic rank coincides with the one of its general entry locus, and show that any smooth and irreducible projective variety admits an embedding with this property.