Abstract
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.
Highlights
In this article, we work over the field of complex numbers
As an extension of the studies of special divisor classes on a del Pezzo surface, we consider the blow-ups of Hirzebruch surfaces
We provide correspondences in [19] that are compatible with the Weyl group W (Fp,r ) = Er+1 between special divisor classes in the Picard group Pic(Fp,r ) and subpolytopes of a Gosset polytope (r − 3)21
Summary
We work over the field of complex numbers. Algebraic surfaces birational to the projective plane are rational surfaces. In [4], the first author showed that the convex hull of Lr in Pic(Sr ) is the Gosset polytope (r − 4) and extends the bijection to correspondences between special divisor classes As an extension of the studies of special divisor classes on a del Pezzo surface, we consider the blow-ups of Hirzebruch surfaces. We extend the results [4] for del Pezzo surfaces to roots, lines, rulings and exceptional systems for blown-up Hirzebruch surfaces. These relations for roots and lines via monoidal transformations are obtained as straightforward comparisons, but rulings on Fp, and exceptional systems on Fp,r involve nontrivial issues To resolve these issues, we consider quartic rational divisor classes, skew a-lines and sextic divisor classes.
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