Fixed Loci of the Anticanonical Complete Linear Systems of Anticanonical Rational Surfaces

Abstract

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 $K_X^2>0$, the following expected result holds: every fixed prime divisor of the complete linear system $|-K_X|$ is a $(-n)$-curve, for some integer $n\ge 1$.