On pluri-half-anticanonical systems of LeBrun twistor spaces
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Abstract:
In this paper, we investigate pluri-half-anticanonical systems on the so-called LeBrun twistor spaces. We determine its dimension, the base locus, the structure of the associated rational map, and also the structure of general members, in precise form. In particular, we show that if $n\ge 3$ and $m\ge 2$, the base locus of the system $|mK^{-1/2}|$ on $n\mathbb {CP}^2$ consists of two non-singular rational curves, along which any member has singularity, and that if we blow up these curves, then the strict transform of a general member of $|mK^{-1/2}|$ becomes an irreducible non-singular surface. We also show that if $n\ge 4$ and $m\ge n-1$, then the last surface is a minimal surface of general type with vanishing irregularity. We also show that the rational map associated to the system $|mK^{-1/2}|$ is birational if and only if $m\ge n-1$.References
- M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no.Β 1711, 425β461. MR 506229, DOI 10.1098/rspa.1978.0143
- N. Honda, Explicit construction of new Moishezon twistor spaces, J. Differential Geom. 82 (2009) 411-444.
- Nobuhiro Honda, Double solid twistor spaces: the case of arbitrary signature, Invent. Math. 174 (2008), no.Β 3, 463β504. MR 2453599, DOI 10.1007/s00222-008-0139-5
- N. Honda, A new series of compact minitwistor spaces and Moishezon twistor spaces over them, to appear in J. Reine Angew. Math., arXiv:0805.0042
- Herbert Kurke, Classification of twistor spaces with a pencil of surfaces of degree $1$. I, Math. Nachr. 158 (1992), 67β85. MR 1235296, DOI 10.1002/mana.19921580105
- Claude LeBrun, Explicit self-dual metrics on $\textbf {C}\textrm {P}_2\#\cdots \#\textbf {C}\textrm {P}_2$, J. Differential Geom. 34 (1991), no.Β 1, 223β253. MR 1114461
- Henrik Pedersen and Yat Sun Poon, Self-duality and differentiable structures on the connected sum of complex projective planes, Proc. Amer. Math. Soc. 121 (1994), no.Β 3, 859β864. MR 1195729, DOI 10.1090/S0002-9939-1994-1195729-1
- Y. Sun Poon, Compact self-dual manifolds with positive scalar curvature, J. Differential Geom. 24 (1986), no.Β 1, 97β132. MR 857378
- Y. Sun Poon, On the algebraic structure of twistor spaces, J. Differential Geom. 36 (1992), no.Β 2, 451β491. MR 1180390
Additional Information
- Nobuhiro Honda
- Affiliation: Department of Mathematics, Tokyo Institute of Technology, O-okayama, Tokyo, Japan
- Email: honda@math.titech.ac.jp
- Received by editor(s): June 29, 2009
- Received by editor(s) in revised form: September 7, 2009, and September 15, 2009
- Published electronically: December 8, 2009
- Additional Notes: The author was partially supported by the Grant-in-Aid for Young Scientists (B), The Ministry of Education, Culture, Sports, Science and Technology, Japan.
- Communicated by: Jon G. Wolfson
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2051-2060
- MSC (2010): Primary 32L25; Secondary 53C28
- DOI: https://doi.org/10.1090/S0002-9939-09-10207-1
- MathSciNet review: 2596041