Slope inequalities for fibred surfaces via GIT

TL;DR

This paper generalizes a theorem linking Geometric Invariant Theory to fibred surface invariants, enabling new bounds and proofs for slope inequalities and related invariants.

Contribution

It extends a theorem by Cornalba and Harris to better analyze invariants of fibred surfaces using GIT, providing new proofs and bounds.

Findings

New proof of the slope inequality

Bound for invariants of double cover fibrations

Generalization of a GIT-based theorem

Abstract

In this paper we present a generalisation of a theorem due to Cornalba and Harris, which is an application of Geometric Invariant Theory to the study of invariants of fibrations. In particular, our generalisation makes it possible to treat the problem of bounding the invariants of general fibred surfaces. As a first application, we give a new proof of the slope inequality and of a bound for the invariants associated to double cover fibrations.