Annali di Matematica Pura ed Applicata (1923 -)

, Volume 198, Issue 2, pp 443–463

# Generically nef vector bundles on ruled surfaces

• Beorchia Valentina
• Zucconi Francesco
Article

## Abstract

The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta–Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov-type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section, see Theorem 3.1. This gives an affirmative answer in this case to a problem posed by Peternell [17]. Concerning ruled surfaces with a negative section, we prove a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor, see Theorem 3.5. Finally, we use such results to bound the invariants of curve fibrations, which factor through finite covers of ruled surfaces.

## Keywords

Vector bundles Chern classes Fibrations Finite covers

14J60 14D06

## Notes

### Acknowledgements

This research is supported by national MIUR funds, PRIN project Geometria delle varietà algebriche (2015). Beorchia Valentina is also supported by national MIUR funds FINANZIAMENTO ANNUALE INDIVIDUALE DELLE ATTIVITÀ BASE DI RICERCA - 2018. Zucconi Francesco is supported by Università degli Studi di Udine - DIMA project Geometry PRIDZUCC2017.

## References

1. 1.
Arbarello, E., Cornalba, M., Griffiths, P.: Geometry of Algebraic Curves. A Series of Comprehensive Studies in Mathemstics 268, vol. II. Springer, Berlin (2011)
2. 2.
Barja, M.A., Stoppino, L.: Stability and singularities of relative hypersurfaces. Int. Math. Res. Not. IMRN 4, 1026–1053 (2016)
3. 3.
Barja, M.A., Stoppino, L.: Positivity properties of relative complete intersections. arXiv:1410.3009 [math.AG] (2014)
4. 4.
Brosius, J.E.: Rank-2 vector bundles on a ruled surface. I. Math. Ann. 265(2), 155–168 (1983)
5. 5.
Beorchia, V., Zucconi, F.: On the slope of fourgonal semistable fibrations. Math. Res. Lett. 25(3), 723–757 (2018)
6. 6.
Casnati, G., Ekedahl, T.: Covers of algebraic varieties. I. A general structure theorem, covers of degree 3, 4 and Enriques surfaces. J. Algebr. Geom. 5(3), 439–460 (1996)
7. 7.
Cornalba, M., Harris, J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. Ec. Norm. Super. 21(4), 455–475 (1988)
8. 8.
Deopurkar, A., Patel, A.: Vector bundles and finite covers, preprint arXiv:1608.01711
9. 9.
Enokizono, M.: Slopes of fibered surfaces with a finite cyclic automorphism. Mich. Math. J. 66(1), 125–154 (2017)
10. 10.
Fedorchuk, M., Jensen, D.: Stability of 2nd Hilbert points of canonical curves. Int. Math. Res. Not. IMRN 2013(22), 5270–5287 (2013)
11. 11.
Lazarsfeld, R.: Nefness modulo the branch locus of the bundle associated to a branched covering. Commun. Algebra 28, 5598–5599 (2000)
12. 12.
Lazarsfeld, R.: Positivity in Algebraic Geometry, II, Positivity for Vector Bundles, and Multiplier Ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 49. Springer, Berlin (2004)
13. 13.
Lu, X., Zuo, K.: On the gonality and the slope of a fibered surface. Adv. Math. 324, 336–354 (2018)
14. 14.
Moriwaki, A.: A sharp slope inequality for general stable fibrations of curves. J. Reine Angew. Math. 480, 177–195 (1996)
15. 15.
Miranda, R.: Triple covers in algebraic geometry. Am. J. Math. 107(5), 1123–1158 (1985)
16. 16.
Nakayama, N.: Zariski-decomposition and abundance, MSJ Memoirs, vol. 14. Mathematical Society of Japan, Tokyo (2004)
17. 17.
Peternell, Th.: Generically nef vector bundles and geometric applications. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds.) Complex and Differential Geometry: Conference held at Leibniz Universität Hannover, September 14–18, 2009, Springer, Berlin (2011)Google Scholar
18. 18.
Peternell, Th, Sommese, A.J.: Ample vector bundles and branched coverings. Commun. Algebra 28(12), 5573–5599 (2000)
19. 19.
Stankova, Z.: Moduli of trigonal curves. J. Algebr. Geom. 9(4), 607–662 (2000)
20. 20.
Xiao, G.: Fibred algebraic surfaces with low slope. Math. Ann. 276, 449–466 (1987)
21. 21.
Viehweg, E.: Die Additivität der Kodaira Dimension für projektive Faserräume über Varietäten des allgemeinen Typs. J. Reine Angew. Math. 330, 132–142 (1982)

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Beorchia Valentina
• 1
• Zucconi Francesco
• 2
1. 1.Dipartimento di Matematica e Geoscienze, Dipartimento di Eccellenza 2018-2020Università di TriesteTriesteItaly
2. 2.Dipartimento di Scienze Matematiche, Informatiche e FisicheUniversità degli studi di UdineUdineItaly