# Strict inclusions of high rank loci

## TL;DR

This paper investigates the structure of high rank loci in projective varieties, providing explicit examples of strict inclusions between these loci and criteria for finiteness of maximal rank points, advancing understanding of rank stratification.

## Contribution

It constructs infinitely many examples of strict inclusions between high rank loci and offers criteria for the finiteness of maximal rank points in space curves.

## Key findings

- Examples of strict inclusion between high rank loci in Veronese surfaces and space curves.
- Criteria for determining finiteness of points with maximal rank 3 in space curves.
- Extension of known examples from curves to higher-dimensional varieties.

## Abstract

For a given projective variety $X$, the high rank loci are the closures of the sets of points whose $X$-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is for the Veronese surface of plane quartics. Although Piene had already shown an example when $X$ is a curve, we construct infinitely many curves in $\mathbb P^4$ for which such strict inclusion appears. For space curves, we give two criteria to check whether the locus of points of maximal rank 3 is finite (possibly empty).