  In this paper we introduce and study divisorial (i) classes} for the blow up
of projective space in several points for i=-1,0 and 1. We generalize Noether's
inequality, and we prove that all divisorial (i) classes are in bijective
correspondence with the orbit of the Weyl group action on one exceptional
divisor following Nagata's original approach. Moreover, we prove that the
irreducibility condition from the definition of divisorial (i) classes can be
replaced by the numerical condition of having positive intersection with all
divisorial (i) classes of smaller degree via the Mukai pairing.

Algebraic Geometry