X-ranks for embedded varieties and extensions of fields

Abstract

Let X ⊂ P r be a projective embedded variety defined over a field K . Results relating maximum and generic X -rank of points of P r ( K ) and P r ( L ) are given, where L is a field containing K . Some of these results are algebraically closed for K and L . In other results (e.g. on the cactus rank), L is a finite extension of K .