Annals of Global Analysis and Geometry

, Volume 39, Issue 3, pp 293–323 | Cite as

Minitwistor spaces, Severi varieties, and Einstein–Weyl structure

Original Paper


In this article, we show that the space of nodal rational 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 of smooth rational 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.


Minitwistor space Twistor space Einstein–Weyl structure Severi variety Nodal rational curve Penrose correspondence 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of Science and EngineeringTokyo Institute of TechnologyMeguroJapan
  2. 2.Mathematical Institute, Graduate School of ScienceTohoku UniversitySendaiJapan
  3. 3.Department of Mathematics, Faculty of Science and TechnologyTokyo University of ScienceChibaJapan

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