平面有理尖点曲线的分类。I. C** 纤维化
Classification of planar rational cuspidal curves. I. C**-fibrations
摘要：
为了对复杂有理尖曲线$E\subseteq \mathbb{P}^2$进行分类，仍然需要对具有对数一般类型的补数的曲线进行分类，即$κ(K_X+D)=2$，其中$(X,D)$是$(\mathbb{P}^2,E)$的对数分辨率。证明了$κ(K_X+\frac{1}{2}D)=-\infty$和$\mathbb{P}^2\setminus E$是$\mathbb{C}^{**}$纤维化的，其中$\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$或$-(K_X+\frac{1}{2}D)$在$(X,\frac{1}{2}D)$的某个最小模型上是充分的.在这里，我们分类，直到一个投影等价，这些有理尖点曲线的补充是$\mathbb{C}^{**}$纤维。从丰富的已知例子列表中，只有极少数不是这种类型。我们还发现了一个新的无限族的双尖曲线与不寻常的性质。
英文摘要：
To classify complex rational cuspidal curves $E\subseteq \mathbb{P}^2$ it remains to classify the ones with complement of log general type, i.e. the ones for which $κ(K_X+D)=2$, where $(X,D)$ is a log resolution of $(\mathbb{P}^2,E)$. It is conjectured that $κ(K_X+\frac{1}{2}D)=-\infty$ and hence $\mathbb{P}^2\setminus E$ is $\mathbb{C}^{**}$-fibered, where $\mathbb{C}^{**}=\mathbb{C}^1\setminus\{0,1\}$, or $-(K_X+\frac{1}{2}D)$ is ample on some minimal model of $(X,\frac{1}{2}D)$. Here we classify, up to a projective equivalence, those rational cuspidal curves for which the complement is $\mathbb{C}^{**}$-fibered. From the rich list of known examples only very few are not of this type. We also discover a new infinite family of bicuspidal curves with unusual properties.
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