Labels of real projective varieties

TL;DR

This paper explores the concept of admissible rank for real projective varieties, analyzing its properties, introducing a scheme-theoretic version, and comparing typical labels with real ranks, with special focus on rational normal curves.

Contribution

It introduces the notion of admissible rank for real projective varieties, studies its properties, and compares it with real and complex ranks, including a scheme-theoretic approach.

Findings

Admissible and complex ranks coincide for rational normal curves.

Introduction of scheme-theoretic admissible rank.

Analysis of typical labels and their differences from real ranks.

Abstract

Let $X$ be a complex projective variety defined over $\mathbb R$. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to $X$. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks.