We consider blowups at a general point of weighted projective planes and, more generally, of toric surfaces with Picard number 1. We give a unifying construction of negative curves on these blowups such that all previously known families appear as boundary cases of this. The classification consists of two classes of said curves, each depending on two parameters. Every curve in these two classes is algebraically related to other curves in both classes; this allows us to find their defining equations inductively. For each curve in our classification, we consider a family of blowups in which the curve defines an extremal class in the effective cone. We give a complete classification of these blowups into Mori Dream Spaces (MDS) and non-MDS. Our approach greatly simplifies previous proofs, avoiding positive characteristic methods and higher cohomology.

中文翻译：

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在加权投影平面的放大中产生极值射线的曲线

我们考虑加权投影平面的一般点处的膨胀，更一般地，考虑皮卡德数为 1 的复曲面的膨胀。我们在这些膨胀上给出了负曲线的统一构造，使得所有先前已知的系列都出现为这种情况的边界情况。分类由两类所述曲线组成，每类取决于两个参数。这两个类别中的每条曲线都与这两个类别中的其他曲线在代数上相关；这使我们能够归纳地找到它们的定义方程。对于我们分类中的每条曲线，我们考虑一系列放大，其中曲线定义了有效锥中的极值类。我们将这些爆炸完全分类为 Mori Dream Spaces (MDS) 和非 MDS。我们的方法大大简化了以前的证明，避免了正特征方法和更高的上同调。