Rational surfaces with anticanonical divisor not reduced

Abstract
We prove the finite generation of the monoid of effective divisor
classes on a smooth projective rational surface X endowed with an anticanonical
divisor such that all its irreducible components are of multiplicity
one except one which has multiplicity two. In almost all cases,
the self-intersection of a canonical divisor KX on X is strictly negative,
hence KX is neither ample nor numerically effective. In particular,
X is not a Del Pezzo surface. Furthermore, it is shown that the first
cohomology group of a numerically effective divisor vanishes; as a consequence,
we determine the dimension of the complete linear system
associated to any given divisor on X.