## Threefolds containing Bordiga surfaces as ample divisors

HTML articles powered by AMS MathViewer

- by Hidetoshi Maeda PDF
- Proc. Amer. Math. Soc.
**137**(2009), 1631-1639 Request permission

## 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 $\mathcal {E}$.## References

- Mauro C. Beltrametti and Andrew J. Sommese,
*The adjunction theory of complex projective varieties*, De Gruyter Expositions in Mathematics, vol. 16, Walter de Gruyter & Co., Berlin, 1995. MR**1318687**, DOI 10.1515/9783110871746 - Takao Fujita,
*Classification theories of polarized varieties*, London Mathematical Society Lecture Note Series, vol. 155, Cambridge University Press, Cambridge, 1990. MR**1162108**, DOI 10.1017/CBO9780511662638 - Paltin Ionescu,
*Embedded projective varieties of small invariants*, Algebraic geometry, Bucharest 1982 (Bucharest, 1982) Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, pp. 142–186. MR**749942**, DOI 10.1007/BFb0071773 - Antonio Lanteri and Hidetoshi Maeda,
*Ample vector bundles and Bordiga surfaces*, Math. Nachr.**280**(2007), no. 3, 302–312. MR**2292152**, DOI 10.1002/mana.200410483 - Antonio Lanteri and Hidetoshi Maeda,
*Projective manifolds of sectional genus three as zero loci of sections of ample vector bundles*, Math. Proc. Cambridge Philos. Soc.**144**(2008), no. 1, 109–118. MR**2388237**, DOI 10.1017/S0305004107000813 - H. Maeda,
*The threefold containing the Bordiga surface of degree ten as a hyperplane section*, Math. Proc. Cambridge Philos. Soc.**145**(2008), 619–622. - Christian Okonek, Michael Schneider, and Heinz Spindler,
*Vector bundles on complex projective spaces*, Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., 1980. MR**561910**

## 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
- Email: maeda@variety.sci.waseda.ac.jp
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**137**(2009), 1631-1639 - MSC (2000): Primary 14J25; Secondary 14J30, 14J60
- DOI: https://doi.org/10.1090/S0002-9939-08-09752-9
- MathSciNet review: 2470821