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We determine the fixed locus of the anticanonical complete linear system of agiven anticanonical rational surface. The case of a geometrically ruledrational surface is fully studied, e.g., the monoid of numerically effectivedivisor classes of such surface is explicitly determined and is minimallygenerated by two elements. On the other hand, as a consequence in theparticular case where $X$ is a smooth rational surface with $K_{X}^{2}>0$, thefollowing expected result holds: every fixed prime divisor of the completelinear system $|-K_X|$ is a $(-n)$-curve, for some integer $n\geq 1$.

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