On the finite generation of the monoid of effective divisor classes on rational surfaces of type (n, m)

Abstract

We prove that every projective rational surface of type (n, m) has only a finite number of (-1) curves and only a finite number of (-2) curves, where n and m are nonnegative integers satisfying the inequality mn-m-4n<0. As a consequence of this result, it follows the finite generation of the monoid of effective divisor classes on such surface. Thus giving in particular, a uniform proof of the classical results stating that the monoid of effective divisor classes on the surface obtained by blowing up the projective plane at points which are either all collinear or which are all on a conic is finitely generated.