关于九尖K3-Surfaces的分类
On the Classification of K3-Surfaces with Nine Cusps
摘要：
所谓有九个尖点的K3-曲面，我指的是有九个孤立的二重点$A_2$的紧致复曲面，但除此之外是光滑的，使得它的最小去奇异化是K3-曲面。在较早的文件中，我表明，每一个这样的表面是一个商的复环面由一个循环群的三阶。在这里，我尝试对这些$K3$ -曲面进行分类，使用复杂环面的周期图。特别是我显示：一个$K3$ -表面与9个尖点进行极化只有0度或2模6。这特别意味着在有九个尖点的射影三维空间中不存在四次曲面。（T.浦部向我指出了如何从尼库林的一个定理推导出这一点。）在附录中给出了三维空间中具有八个尖点的四次曲面的显式方程。
英文摘要：
By a K3-surface with nine cusps I mean a compact complex surface with nine isolated double points $A_2$, but otherwise smooth, such that its minimal desingularisation is a K3-surface. In an earlier paper I showd that each such surface is a quotient of a complex torus by a cyclic group of order three. Here I try to classify these $K3$-surfaces, using the period map for complex tori. In particular I show: A $K3$-surface with nine cusps carries polarizations only of degrees 0 or 2 modulo 6. This implies in particular that there is no quartic surface in projective three-space with nine cusps. (T. Urabe pointed out to me how to deduce this from a theorem of Nikulin.) In an appendix I give explicit equations of quartic surfaces in three-space with eight cusps.
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