Abstract
We construct complex surfaces with genus two fibrations over \({\mathbb P^1}\) having special fibres such that the minimum of the multiplicities of the components is ≥ 2 whereas the g.c.d is 1. We can then produce new examples of fibred surfaces without multiple fibres which are of “general type” according to the definition of Campana. We prove that these surfaces are of general type and simply connected; and we compute in some cases their invariants. Moreover, we extend the construction obtaining general type fibrations of any even genus on simply connected surfaces. All our examples are defined over number fields.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramovich, D.: Birational geometry for number theorists, Arithmetic geometry. Clay Math. Proc. 8, 335–373, Am. Math. Soc. (2009)
Barth W.P., Hulek K., Peters C.A.M., Van de Ven A.: Compact Complex Surfaces. 2nd edn. Springer, Berlin, Heidelberg (2004)
Beauville A.: Le nombre minimum de fibres singulieres d’une courbe stable sur \({\mathbb{P}^1}\). Asterisque 86, 97–108 (1981)
Campana F.: Fibres multiples sur les surfaces: aspects geométriques, hyperboliques et arithmetiques. Manuscr. Math. 117, 429–461 (2005)
Campana F.: Negativity of compact curves in infinite covers of projective surfaces. J. Algebraic Geom. 7(4), 673–693 (1998)
Campana F.: Orbifolds, special varieties and classification theory. Ann. Inst. Fourier (Grenoble) 54(3), 499–630 (2004)
Campana, F.: Special orbifolds and birational classification: a survey, arXiv:1001.3763v1 [math.AG]
Colliot-Thélène J.-L., Skorobogatov A.N., Swinnerton-Dyer P.: Double fibres and double covers: paucity of rational points. Acta Arith. 79(2), 113–135 (1997)
Horikawa E.: On deformations of quintic surfaces. Invent. Math. 31, 43–85 (1975)
Namikawa Y., Ueno K.: The complete classification of fibres in pencils of curves of genus two. Manuscr. Math. 9, 143–186 (1973)
Ogg A.P.: On pencils of curves of genus two. Topology 5, 355–362 (1966)
Persson U.: Chern invariants of surfaces of general type. Compos. Math. 43, 53–58 (1982)
Tan S.L.: The minimal number of singular fibers of a semistable curve over \({\mathbb{P}^1}\). J. Algebraic Geom. 4(3), 591–596 (1995)
Tan S.L., Tu Y., Zamora A.G.: On complex surfaces with 5 or 6 semistable singular fibers over \({\mathbb{P}^1}\). Math. Z. 249, 427–438 (2005)
Winters G.B.: On the existence of certain families of curves. Am. J. Math. 96(2), 215–228 (1974)
Xiao G.: Hyperelliptic surfaces of general type with K 2 < 4χ. Manuscr. Math. 57(2), 125–148 (1987)
Xiao G.: Surfaces fibrées en courbes de genre deux, LNM, vol. 1137. Springer, Berlin (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stoppino, L. Fibrations of Campana general type on surfaces. Geom Dedicata 155, 69–80 (2011). https://doi.org/10.1007/s10711-011-9578-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-011-9578-z