Partially complex ranks for real projective varieties

TL;DR

This paper investigates the existence of small, conjugation-invariant subsets of complex projective varieties that contain a given real point in their span, with applications to real ranks and specific varieties like hypersurfaces and Veronese varieties.

Contribution

It introduces the concept of partially complex ranks for real projective varieties and analyzes their properties and advantages over traditional real ranks.

Findings

Existence of conjugation-invariant sets with small cardinality for real points.

Comparison between these additive decompositions and the real rank.

Specific results for hypersurfaces and Veronese varieties.

Abstract

Let $X(\mathbb {C})\subset \mathbb {P}^r(\mathbb {C})$ be an integral non-degenerate variety defined over $\mathbb {R}$. For any $q\in \mathbb {P}^r(\mathbb {R})$ we study the existence of $S\subset X(\mathbb {C})$ with small cardinality, invariant for the complex conjugation and with $q$ contained in the real linear space spanned by $S$. We discuss the advantages of these additive decompositions with respect to the $X(\mathbb {R})$-rank, i.e. the rank of $q$ with respect to $X(\mathbb {R})$. We describe the case of hypersurfaces and Veronese varieties.