Slope inequality for an arbitrary divisor

TL;DR

This paper generalizes the slope inequality for algebraic surfaces by considering arbitrary divisors and sub-sheaves, analyzing how their properties influence the inequality and providing examples and applications.

Contribution

It introduces a broader slope inequality framework for arbitrary divisors and sub-sheaves on surfaces, extending previous results and exploring the role of divisor speciality.

Findings

Generalized slope inequality for arbitrary divisors

Role of divisor speciality in the inequality

Examples and applications demonstrating the theory

Abstract

Let $f: S \longrightarrow C$ be a surjective morphism with connected fibers from a smooth complex projective surface $S$ to a smooth complex projective curve $C$ with general fiber $F$. In this paper, we develop a more general version of the slope inequality for data $(D, \mathcal{F})$, where $D$ is an arbitrary relatively effective divisor on $S$ and $\mathcal{F}$ is a locally free sub-sheaf of $f_{*}\mathcal{O}_{S}(D)$. We see how the speciality of $D$, restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.