On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
Toru Tsukioka
Abstract
We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in the product of a projective space and a projective line. We give a classification result for a special case. We determine the nef cones and describe the extremal contractions for related smooth projective varieties.
Acknowledgements
The author would like to thank Kazunori Yasutake for helpful comments on weak Fano manifolds, and Kento Fujita for several observations on the birational geometry of algebraic varieties concerning Theorem 1.1. The author would also like to thank the referee for many suggestions, which enabled the author to simplify the proof of Theorem 1.1 and to describe explicitly the extremal contractions. This work was supported by JSPS KAKENHI Grant Number JP23740028.
Communicated by: I. Coskun
References
[1] L. Bonavero, F. Campana, J. A. Wiśniewski, Variétés complexes dont l’éclatée en un point est de Fano. C. R. Math. Acad. Sci. Paris 334 (2002), 463–468. MR1890634 Zbl 1036.14020Search in Google Scholar
[2] C. Casagrande, On the birational geometry of Fano 4-folds. Math. Ann. 355 (2013), 585–628. MR3010140 Zbl 1260.14052Search in Google Scholar
[3] C. Casagrande, Fano 4-folds with rational fibrations. Algebra Number Theory 14 (2020), 787–813. MR4113781 Zbl 07213915Search in Google Scholar
[4] O. Debarre, Higher-dimensional algebraic geometry. Springer 2001. MR1841091 Zbl 0978.14001Search in Google Scholar
[5] W. Fulton, Intersection theory. Springer 1998. MR1644323 Zbl 0885.14002Search in Google Scholar
[6] V. A. Iskovskikh, Fano 3-folds. I. Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 516–562, 717. MR463151 Zbl 0382.14013Search in Google Scholar
[7] V. A. Iskovskikh, Fano 3-folds. II. Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 506–549. MR503430 Zbl 0424.14012Search in Google Scholar
[8] V. A. Iskovskikh, Y. G. Prokhorov, Fano varieties. In: Algebraic geometry, V, volume 47 of Encyclopaedia Math. Sci., 1–247, Springer 1999. MR1668579 Zbl 0912.14013Search in Google Scholar
[9] Y. Kawamata, Small contractions of four-dimensional algebraic manifolds. Math. Ann. 284 (1989), 595–600. MR1006374 Zbl 0661.14009Search in Google Scholar
[10] S. L. Kleiman, Toward a numerical theory of ampleness. Ann. of Math. (2) 84 (1966), 293–344. MR206009 Zbl 0146.17001Search in Google Scholar
[11] J. Kollár, S. Mori, Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge Univ. Press 1998. MR1658959 Zbl 0926.14003Search in Google Scholar
[12] R. Lazarsfeld, Positivity in algebraic geometry. I. Springer 2004. MR2095471 Zbl 1093.14501Search in Google Scholar
[13] S. Mori, S. Mukai, Classification of Fano 3-folds with B2 ≥ 2. Manuscripta Math. 36 (1981/82), 147–162. MR641971 Zbl 0478.14033Search in Google Scholar
[14] J. C. Ottem, Birational geometry of hypersurfaces in products of projective spaces. Math. Z. 280 (2015), 135–148. MR3343900 Zbl 1333.14013Search in Google Scholar
[15] J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds. J. Reine Angew. Math. 417 (1991), 141–157. MR1103910 Zbl 0721.14023Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
