  The aim of this note is to shed some light on the relationships among some
notions of positivity for vector bundles that arose in recent decades.
  Our purpose is to study several of the positivity notions studied for vector
bundles with some notions of asymptotic base loci that can be defined on the
variety itself, rather than on the projectivization of the given vector bundle.
We relate some of the different notions conjectured to be equivalent with the
help of these base loci, and we show that these can help simplify the various
relationships between the positivity properties present in the literature.
  In particular, we define augmented and restricted base loci $\mathbb{B}_+
(E)$ and $\mathbb{B}_- (E)$ of a vector bundle $E$ on the variety $X$, as
generalizations of the corresponding notions studied extensively for line
bundles. As it turns out, the asymptotic base loci defined here behave well
with respect to the natural map induced by the projectivization of the vector
bundle $E$.
