Minitwistor spaces, Severi varieties, and Einstein–Weyl structure
Minitwistor spaces, Severi varieties, and Einstein–Weyl structure
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DOI：
10.1007/s10455-010-9235-z
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发表时间：
2009-01
影响因子：
0.7
通讯作者：
N. Honda;Fuminori Nakata
中科院分区：
文献类型：
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作者：
N. Honda;Fuminori Nakata
本文证明了任意非奇异射影曲面上的结点有理曲线空间，即所谓的(有理曲线的)Severi簇，如果是三维的，则总是具有自然的Einstein-Weyl结构。这是Hitchin在复曲面上光滑有理曲线空间上Einstein-Weyl结构的推广。作为与Einstein-Weyl结构密切相关的几何对象，我们研究了Severi簇上的零曲面和测地线。我们还看到，如果射影曲面具有适当的实结构，则Severi簇的实轨迹成为正定的爱因斯坦-韦尔流形。此外，我们还构造了具有三维Severi有理曲线的有理曲面的各种显式例子。
In this article, we show that the space ofnodalrational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein–Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein–Weyl structure on the space ofsmoothrational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein–Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein–Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves.