Blown-up Hirzebruch surfaces and special divisor classes

We work on special divisor classes on blow-ups F_p,r of Hirzebruch surfaces over the field of complex numbers, and extend fundamental properties of special divisor classes on del Pezzo surfaces parallel to analogous ones on surfaces F_p,r. We also consider special divisor classes on surfaces F_p,r with respect to monoidal transformations and explain the tie-ups among them contrast to the special divisor classes on del Pezzo surfaces. In particular, the fundamental properties of quartic rational divisor classes on surfaces F_p,r are studied, and we obtain interwinded relationships among rulings, exceptional systems and quartic rational divisor classes along with monoidal transformations. We also obtain the effectiveness for the rational divisor classes on F_p,r with positivity condition.