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Minitwistor spaces, Severi varieties, and Einstein-Weyl structure. (English) Zbl 1222.53053

Consider a non-singular complex projective surface \(Z\) and a smooth rational curve \(Y\subset Z\). The self-intersection number of \(Y\) is 2 if and only if the deformations of \(Y\) inside \(Z\) are unobstructed and the deformation space is 3-dimensional. In this case, N. J. Hitchin [Lect. Notes Math. 970, 79–99 (1982; Zbl 0507.53025)] proved that the deformation space admits an Einstein-Weyl structure (in the holomorphic category).
In the present paper, the authors develop a similar theory for the Severi variety of rational curves, i.e., the space of nodal (instead of smooth) rational curves on a non-singular complex projective surface.
If the surface has an appropriate real structure, then the real locus of the Severi variety turns out to be a positive definite Einstein-Weyl manifold.

MSC:

53C28 Twistor methods in differential geometry
32Q25 Calabi-Yau theory (complex-analytic aspects)

Citations:

Zbl 0507.53025
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Full Text: DOI arXiv

References:

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