Irreducibility of −1-classes on anticanonical rational surfaces and finite generation of the effective monoid

Abstract

We give necessary and sufficient conditions for a divisor class on smooth projective anticanonical rational surfaces to be the class of a smooth rational curve of self-intersection -1. We characterize smooth projective anticanonical rational surfaces for which the monoid of classes (modulo algebraic equivalence) of effective divisors is not ?nitely generated, extending results of Lahyane for the case of rational surfaces X with KX2 = 0.