Typical and admissible ranks over fields

Abstract

Let $X(\mathbb {R})$ be a geometrically connected variety defined over $\mathbb {R}$ such that the set of all its complex points $X(\mathbb {C})$ is non-degenerate. We introduce the notion of admissible rank of a point P with respect to X to be the minimal cardinality of a set $S\subset X(\mathbb {C})$ of points such that S spans P and S is stable under conjugation. Any set evincing the admissible rank can be equipped with a label keeping track of the number of its complex and real points. We show that, in the case of generic identifiability, there is an open dense euclidean subset of points with certain admissible rank for any possible label. Moreover we show that if X is a rational normal curve then there always exists a label for the generic element. We present two examples in which either the label doesn’t exist or the admissible rank is strictly bigger than the usual complex rank.