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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
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