On divisorial (i) classes

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 (-1) classes of smaller degree via the Mukai pairing.