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On pluri-half-anticanonical systems of LeBrun twistor spaces

Author: Nobuhiro Honda
Journal: Proc. Amer. Math. Soc. 138 (2010), 2051-2060
MSC (2010): Primary 32L25; Secondary 53C28
Published electronically: December 8, 2009
MathSciNet review: 2596041
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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 $m\ge n-1$.

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Nobuhiro Honda
Affiliation: Department of Mathematics, Tokyo Institute of Technology, O-okayama, Tokyo, Japan

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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.