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Threefolds containing Bordiga surfaces as ample divisors

Author: Hidetoshi Maeda
Journal: Proc. Amer. Math. Soc. 137 (2009), 1631-1639
MSC (2000): Primary 14J25; Secondary 14J30, 14J60
Published electronically: November 26, 2008
MathSciNet review: 2470821
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Abstract: Let $ L$ be an ample line bundle on a smooth complex projective variety $ X$ of dimension three such that there exists a smooth member $ Z$ of $ \vert L\vert $. When the restriction $ L_{Z}$ of $ L$ to $ Z$ is very ample and $ (Z,L_{Z})$ is a Bordiga surface, it is proved that there exists an ample vector bundle $ \mathcal{E}$ of rank two on $ \mathbb{P}^{2}$ with $ c_{1}(\mathcal{E}) = 4$ and $ 3 \leq c_{2}(\mathcal{E}) \leq 10$ such that $ (X,L) = (\mathbb{P}_{\mathbb{P}^{2}}(\mathcal{E}),H(\mathcal{E}))$, where $ H(\mathcal{E})$ is the tautological line bundle on the projective space bundle $ \mathbb{P}_{\mathbb{P}^{2}}(\mathcal{E})$ associated to $ \mathcal{E}$.

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Additional Information

Hidetoshi Maeda
Affiliation: Department of Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

Keywords: Ample line bundle, Bordiga surface
Received by editor(s): November 23, 2007
Received by editor(s) in revised form: July 16, 2008
Published electronically: November 26, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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