The maximum number of singular points on rational homology projective planes
The maximum number of singular points on rational homology projective planes
复制标题
有理同调射影平面上奇异点的最大数量
DOI：
10.1090/s1056-3911-10-00532-1
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
J. Keum
中科院分区：
文献类型：
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作者：
DongSeon Hwang;J. Keum
一个正规射影复曲面称为有理同调射影平面，如果它与复射影平面$\mathbb{C}\mathbb{P}^2 $有相同的贝蒂数。已知具有商奇点的有理同调射影平面至多有5个奇点。到目前为止，所有已知的例子最多有4个奇点。本文证明了具有商奇点的有理同调射影平面S使得K_S是nef，除一种情况外，至多有4个奇点。例外情况来自Enriques曲面的9个光滑有理曲线的配置，其Dynkin图的类型为3A_1 \oplus 2A_3$。
我们也得到了类似的结果，在可微的情况下，在辛的情况下，在一定的假设下，所有举行的代数情况。
A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $\mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. So far all known examples have at most 4 singular points. In this paper, we prove that a rational homology projective plane $S$ with quotient singularities such that $K_S$ is nef has at most 4 singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type $ 3A_1 \oplus 2A_3$.
We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case.