论文标题
具有理性同源磁盘平滑的复杂表面奇异性
Complex surface singularities with rational homology disk smoothings
论文作者
论文摘要
$ p^2/pq-1 $的环状商奇异性（$ 0 <q <p，（p，q）= 1 $）具有平滑性，其Milnor纤维为$ \ Mathbb Q $ HD或理性同源性磁盘（即Milnor编号为$ 0 $）（[9]，5.9.1）。在1980年代，我们发现了此类奇异性的其他例子：三个三级侵入和六个单一无限的家庭，都是加权同质的。后来的Stipsicz，Szabó，Bhupal和作者（[7]，[1]）的工作证明了这些是唯一的加权同质例子。在他的UNC博士学位论文（未发表但在[2]上获得）中，我们的学生雅各布·福勒（Jacob Fowler）完成了这些奇点的分析分类，并计算了每种情况下的平滑次数，除了$ \ Mathcal W $，$ \ Mathcal n $和$ \ \ \ \ Mathcal M $以外的类型。在本文中，我们描述了他的结果，并解决了剩余的案例；除了明显的对称性二元图，有一个唯一的$ \ mathbb Q $ HD平滑组件。该方法涉及研究在投影有理表面上有理曲线的配置。
A cyclic quotient singularity of type $p^2/pq-1$ ($0<q<p, (p,q)=1$) has a smoothing whose Milnor fibre is a $\mathbb Q$HD, or rational homology disk (i.e., the Milnor number is $0$) ([9], 5.9.1). In the 1980's, we discovered additional examples of such singularities: three triply-infinite and six singly-infinite families, all weighted homogeneous. Later work of Stipsicz, Szabó, Bhupal, and the author ([7], [1]) proved that these were the only weighted homogeneous examples. In his UNC PhD thesis (unpublished but available at [2]), our student Jacob Fowler completed the analytic classification of these singularities, and counted the number of smoothings in each case, except for types $\mathcal W$, $\mathcal N$, and $\mathcal M$. In this paper, we describe his results, and settle these remaining cases; there is a unique $\mathbb Q$HD smoothing component except in the cases of an obvious symmetry of the resolution dual graph. The method involves study of configurations of rational curves on projective rational surfaces.
