# On divisorial (i) classes

## TL;DR

This paper introduces divisorial (i) classes for blow-ups of projective space, generalizes Noether's inequality, and establishes a correspondence with Weyl group orbits, providing new insights into their structure.

## Contribution

It defines divisorial (i) classes for specific blow-ups, generalizes classical inequalities, and links these classes to Weyl group actions, advancing understanding of their algebraic and geometric properties.

## Key findings

- Divisorial (i) classes are in bijection with Weyl group orbits.
- The irreducibility condition can be replaced by a numerical positivity condition.
- Generalization of Noether's inequality to these classes.

## Abstract

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.