Slope inequality of fibered surfaces and moduli of curves

Using the theory of moduli of curves, we establish various slope inequalities for general fibered surfaces under the assumption that every fiber of the relative canonical model is reduced. To prove this, we introduce the notion of functorial divisors on Artin stacks and prove a theorem concerning their effectiveness. Furthermore, we consider the moduli stack of reduced and local complete intersection canonically polarized curves and prove that the closed substack parametrizing non-stable curves has no divisorial components. Applying the two theorems to effective tautological divisors on this moduli stack, we obtain several slope inequalities. As an application, we provide a positive partial answer to the question posed by Lu and Tan regarding the Chern invariants of fiber germs.Comment: 29 page