Relative And Modular Invariants Of Fibrations Of Surfaces

The main purpose of this paper is to study the relative invariants of fibrations, and lower bounds of the modular invariants.For hyperelliptic fibrations, G. Xiao got lower and upper bounds of their slopes. It’s known that there are fibrations with the lowest slope. In1992, Xiao proposed an open problem on the existence of such fibrations with the highest slope for g≥2. One of our purpose is to solve this problem. We construct examples of fibrations of genus g with the highest slope for any g≥2. We get a geometric meaning of Xiao’s singularity indexes. We prove that for a semistable fiber, the singularity index equals to the local intersection number of the corresponding boundary divisor with the fibration.There are three fundamental divisor classes in the moduli space Μg:Hodge divisor class λ, boundary class δ=δ0+…+δ[g/2] and κ=12λ-δ. Hence we have three foundamental modular invariants λ(f), δ(f), and κ(f) respectively. The boundary divisors of moduli space Ηg of hyperelliptic curves are δ0,…,δ0,…, δ[g/2] and ξ0,…,ξ[(g-1)/2]·Suppose f:S→C is a genus g fibration of variable moduli. If g=2, then we show that Furthermore, if one of the inequalities becomes an equality and Xf=2, then there is exactly one family of such curves. Therefore, the lower bounds are optimal. If g≥3and the semistable model of f is not smooth, then we getSuppose f is hyperelliptic. When g=3, we have When g≥4, we haveIn order to prove these inequalities, we get the lower bounds on the local intersection numbers of the boundary divisors δi,ξj with the fibration. When g=2and3, our lower bounds are the best.In the study of effective geometric Bogomolov conjecture, one needs to give lower bound on some kind of height a’(D). Recently, other authors obtain some lower bounds depending on the fibration f. As an application of our result, for g=2,3,4we obtained three absolute lower bounds which depend only on the genus g,...