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

TL;DR

This paper establishes new slope inequalities for fibered surfaces using moduli of curves, introduces functorial divisors on Artin stacks, and applies these results to prove conjectures and inequalities related to the geometry of curves and fibered surfaces.

Contribution

It introduces the notion of functorial divisors on Artin stacks and proves their effectiveness, leading to new slope inequalities and applications to longstanding conjectures.

Findings

Proved a theorem on the effectiveness of functorial divisors on Artin stacks.

Established several slope (in)equalities for fibered surfaces.

Provided positive answers to Reid's conjecture and partial answers to Lu and Tan's question.

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. 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 above theorems to some tautological divisors on various moduli stacks of curves, 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 $3$ and a positive partial answer to the question posed by Lu and Tan regarding the Chern invariants of fiber germs.