TY - JOUR

T1 - Minitwistor spaces, Severi varieties, and Einstein-Weyl structure

AU - Honda, Nobuhiro

AU - Nakata, Fuminori

N1 - Funding Information:
Acknowledgments The authors are partially supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.

PY - 2011/3

Y1 - 2011/3

N2 - 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.

AB - 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.

KW - Einstein-Weyl structure

KW - Minitwistor space

KW - Nodal rational curve

KW - Penrose correspondence

KW - Severi variety

KW - Twistor space

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U2 - 10.1007/s10455-010-9235-z

DO - 10.1007/s10455-010-9235-z

M3 - Article

AN - SCOPUS:79551526769

VL - 39

SP - 293

EP - 323

JO - Annals of Global Analysis and Geometry

JF - Annals of Global Analysis and Geometry

SN - 0232-704X

IS - 3

ER -