Kummer surfaces for the self-product of the cuspidal rational curve
Kummer surfaces for the self-product of the cuspidal rational curve
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尖有理曲线自积的 Kummer 曲面
DOI：
10.1090/s1056-3911-06-00438-3
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发表时间：
2005
影响因子：
1.8
通讯作者：
S. Schroeer
中科院分区：
文献类型：
--
作者：
S. Schroeer
经典的Kummer结构附着在阿贝尔曲面上，即K3曲面。正如Shioda和Katsura所展示的那样，对于超奇异的阿贝尔表面，这种结构被分解为特征二。用有理尖点曲线的自积代替超奇异阿贝尔曲面，用适当的无穷小群格式作用量代替符号对合，针对这种情况给出了正确的Kummer型构造。我们遇到的是D4和D8类型的有理重合点，而不是A1类型。结果表明，生成的曲面是具有Artin不变量1和2的超奇异K3曲面。它们位于通过同时分辨获得的一维族中，该族在纯不可分离的碱基改变后存在。
The classical Kummer construction attaches to an abelian surface a K3 surface. As Shioda and Katsura showed, this construction breaks down for supersingular abelian surfaces in characteristic two. Replacing supersingular abelian surfaces by the selfproduct of the rational cuspidal curve, and the sign involution by suitable infinitesimal group scheme actions, I give the correct Kummer-type construction for this situation. We encounter rational double points of type D4 and D8, instead of type A1. It turns out that the resulting surfaces are supersingular K3 surfaces with Artin invariant one and two. They lie in a 1-dimensional family obtained by simultaneous resolution, which exists after purely inseparable base change.