Modular invariants and singularity indices of hyperelliptic fibrations

Abstract

The modular invariants of a family of curves are the degrees of the pullback of the corresponding divisors by the moduli map. The singularity indices were introduced by Xiao (1991) to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In semistable case, the author shows that the modular invariants corresponding to the boundary divisor classes are just the singularity indices. As an application, the author shows that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in semistable case.

Keywords

Modular invariants / Singularity indices / Moduli space of curves

References

[1] Arbarello E., Cornalba M., Griffiths P. A.. Geometry of Algebraic Curves, 2011, Berlin: Springer-Verlag

[2] Bath W., Peters C., Van de Ven A.. Compact Complex Surfaces, 1984, Berlin: Springer-Verlag

[3] Chen, Z., Lu, J. and Tan, S.-L., On the modular invariants of a family of non-hyperelliptic curves of genus 3, http://math.ecnu.edu.cn/preprint/2010-006.pdf

[4] Cornalba M., Harris J.. Divisor classes associated to families of stable varieties. with applications to the moduli space of curves, Ann. Scient. Ec. Norm. Sup., 1988, 21: 455-475

[5] Horikawa E.. On algebraic surfaces with pencils of curves of genus 2, Complex Analysis and Algebraic Geometry, 1977, London: Cambridge Univ. Press 79-90

[6] Kodaira K.. On compact analytic surfaces. III, Ann. Math., 1963, 78(1): 1-40

[7] Liu X. L., Tan S. L.. Families of hyperelliptic curves with maximal slopes. Science China Mathematics, 2013, 56(9): 1743-1750

[8] Namikawa Y., Ueno K.. The complete classification of fibres in pencils of curves of genus two. Manuscripta Math., 1973, 9: 143-186

[9] Ogg A. P.. On pencils of curves of genus two. Topology, 1966, 5: 355-362

[10] Tan S. L.. Chern numbers of a singular fiber, modular invariants and isotrivial families of curves. Acta Math. Viet., 2010, 35(1): 159-172

[11] Tan, S. L., Chern numbers of holomorphic foliations and Poincaré problems, in preprint.

[12] Tu Y.. A criterion of semi-stable hyperelliptic fibration. Journal of Sichuan University (Natural Science Edition), 2008, 45(5): 1049-1050

[13] Xiao G.. p1 of elliptic and hyperelliptic surfaces. Internat. J. Math., 1991, 2(5): 599-615

[14] Xiao G.. The fibrations of algbraic surfaces, 1992, Shanghai: Shanghai Scientific & Technical Publishers

[15] Zhang S.. Admissible pairing on a curve. Invent. Math., 1993, 112(1): 171-193

[16] Zhang S.. Gross-Schoen cycles and dualising sheaves. Invent. Math., 2010, 179(1): 1-73