  A determination of the fixed components, base points and irregularity is made
for arbitrary numerically effective divisors on any smooth projective rational
surface having an effective anticanonical divisor. All of the results are
proven over an algebraically closed field of arbitrary characteristic.
Applications, treated in separate papers, include questions involving: points
in good position, birational models of rational surfaces in projective space,
and resolutions for ideals of fat point subschemes of $P^2$.
