Fixed loci of the Anticanonical linear systems of Anticanonical Rational Surfaces

We determine the fixed locus of the anticanonical complete
linear system of a given anticanonical rational surface. The case of a
geometrically ruled rational surface is fully studied, e.g., the monoid of
numerically effective divisor classes of such surface is explicitly determined
and is minimally generated by two elements. On the other hand, as a
consequence in the particular case where X is a smooth rational surface
with K^2_X > 0, the following expected result holds: every fixed prime
divisor of the complete linear system |-K_X| is a (-n)-curve, for some
integer n>=1.