## Osaka Journal of Mathematics

### Slope inequalities for fibred surfaces via GIT

Lidia Stoppino

#### Abstract

In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.

#### Article information

Source
Osaka J. Math., Volume 45, Number 4 (2008), 1027-1041.

Dates
First available in Project Euclid: 26 November 2008

https://projecteuclid.org/euclid.ojm/1227708831

Mathematical Reviews number (MathSciNet)
MR2493968

Zentralblatt MATH identifier
1160.14013

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 14J29: Surfaces of general type 14D06: Fibrations, degenerations

#### Citation

Stoppino, Lidia. Slope inequalities for fibred surfaces via GIT. Osaka J. Math. 45 (2008), no. 4, 1027--1041. https://projecteuclid.org/euclid.ojm/1227708831

#### References

• E. Arbarello, M. Cornalba, P.A. Griffiths and J. Harris: Geometry of algebraic curves, Vol. II, in preparation.
• T. Ashikaga and K. Konno: Global and local properties of pencils of algebraic curves; in Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. in Pure Math. 36, Math. Soc. Japan, Tokyo, 2002, 1--49.
• M.A. Barja: On the slope and geography of fibred surfaces and threefolds, Ph.D. Thesis, Univesity of Barcelona, 1998.
• M.A. Barja and L. Stoppino: Linear stability of projected canonical curves with applications to the slope of fibred surfaces, J. Math. Soc. Japan 60 (2008), 171--192.
• M.A. Barja and L. Stoppino: A sharp bound for the slope of general trigonal fibrations of even genus, in preparation.
• M.A. Barja and F. Zucconi: On the slope of fibred surfaces, Nagoya Math. J. 164 (2001), 103--131.
• W.P. Barth, K. Hulek, C.A.M. Peters and A. Van de Ven: Compact Complex Surfaces, second edition, Springer-Verlag, Berlin Heidelberg, 2004.
• J. Bost: Semi-stability and heights of cycles, Invent. Math. 118 (1994), 223--253.
• M. Cornalba and J. Harris: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sc. Ec. Norm. Sup. 21 (1988), 455--475.
• M. Cornalba and L. Stoppino: A sharp bound for the slope of double cover fibrations, to appear in the Michigan Math. J., preprint math.AG/0510144.
• W. Fulton: Intersection Theory, second edition, Springer-Verlag, 1998.
• A. Gibney, S. Keel and I. Morrison: Towards the ample cone of $\bar{M}_{g,n}$, J. Amer. Math. Soc. 15 (2002), 273--294.
• Y. Kawamata, K. Matsuda and K. Matsuki: Introduction to the minimal model problem, Adv. Stud. in Pure Math. 10 (1987), 283--360.
• G.R. Kempf: Instability in invariant theory, Ann. of Math. 108 (1978), 299--316.
• K. Konno: Non-hyperelliptic fibrations of small genus and certain irregular canonical surfaces, Ann. Sc. Norm. Sup. Pisa ser. IV 20 (1993), 575--595.
• A. Moriwaki: A sharp slope inequality for general stable fibrations of curves, J. Reine Angew. Math. 480 (1996), 177--195.
• D. Mumford: Stability of projective varieties, L'Ens. Math. 23 (1977), 39--110.
• R. Pardini: The severi inequality $K^{2}\geq 4\chi$ for surfaces of maximal Albanese dimension, Invent. Math. 159 (2005), 669--672.
• U. Persson: Double coverings and surfaces of general type; in Algebraic geometry (Proc. Sympos., Univ. Troms\o, Troms\o, 1977), Lecture Notes in Math. 687, 1978, 168--195.
• L. Stoppino: Stability of maps to projective spaces, with applications to the slope of fibred surfaces, Ph.D. Thesis, Università di Pavia, 2005.
• S. Tan: On the invariants of base changes of pencils of curves I, Manuscripta Math. 84 (1994), 225-244.
• S. Tan: On the invariants of base changes of pencils of curves II, Math. Z. 222 (1996), 655--676.
• G. Xiao: Fibred algebraic surfaces with low slope, Math. Ann. 276 (1987), 449--466.