Slope inequality of fibered surfaces, Morsification conjecture and moduli of curves

Abstract

Using the theory of moduli of curves, we establish various slope inequalities for general fibered surfaces. More precisely, we introduce the notion of functorial divisors on Artin stacks and prove a theorem concerning their effectiveness. Considering the above concept with the Morsification conjecture and the semistable reduction, we obtain several slope (in)equalities, e.g., a generalization of Moriwaki's slope inequality, slope equalities of general fibered surfaces whose fibers satisfy the Morsification conjecture. As applications, we provide a positive answer to Reid's conjecture concerning algebraic Morsification of non-hyperelliptic fibrations of genus 3, a positive partial answer to the question posed by Lu and Tan regarding the Chern invariants of fiber germs, and a partial result concerning lower bounds of the slope of effective divisors on the moduli spaces of stable curves.