We work on special divisor classes on blow-ups ${\mathbf{F}}_{p,r}$ 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 ${\mathbf{F}}_{p,r}$ . We also consider special divisor classes on surfaces ${\mathbf{F}}_{p,r}$ 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 ${\mathbf{F}}_{p,r}$ are studied, and we obtain interwinded relationships among rulings, exceptional systems and quartic rational divisor classes along with monoidal transformations.

中文翻译：

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Hirzebruch爆破曲面和特殊除数类

我们致力于爆破的特殊除数类 ${\mathbf{F}}_{p，\mathrm{[R}}$ Hirzebruch曲面在复数场上的分布，并在del Pezzo曲面上扩展了特殊除数类的基本性质，使其平行于相似的曲面 ${\mathbf{F}}_{p，\mathrm{[R}}$ 。我们还考虑曲面上的特殊除数类 ${\mathbf{F}}_{p，\mathrm{[R}}$ 关于单曲面变换，并解释其中的联系与del Pezzo曲面上的特殊除数类形成对比。特别是曲面上四次有理除数类的基本性质 ${\mathbf{F}}_{p，\mathrm{[R}}$ 经过研究，我们获得了裁定，例外系统和四次有理数除数类之间的缠绕关系，以及单调变换。