关于复数和四元数的Sylvester-Gallai定理
Sylvester-Gallai Theorems for Complex Numbers and Quaternions
摘要：
Sylvester-Gallai（SG）配置是点的有限集S，使得通过S中任何两个点的线包含S的第三个点。根据Sylvester-Gallai定理，实射影空间中的SG构形必须共线。Serre（1966）的一个问题是问在复射影空间中的SG构形是否必须共面。Kelly（1986）利用Hirzebruch的一个深不等式证明了这一点。我们给出了这个结果的一个初等证明，并将其推广到四元数上的射影空间中的SG位形必包含在一个三维平面中。
英文摘要：
A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.
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