Relative Noether inequality on fibered surfaces

TL;DR

This paper establishes effective bounds on sections of line bundles on fibered surfaces, providing a relative Noether inequality, a new proof of the slope inequality, and insights into the geography of surfaces in positive characteristic.

Contribution

It introduces a relative Noether inequality for fibered surfaces and offers a novel proof of the slope inequality without stability methods.

Findings

Effective upper bounds on global sections of nef line bundles.

Validation of Severi inequality for surfaces of general type in positive characteristic.

Analysis of the geography of surfaces with Albanese fibrations.

Abstract

We prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic. It can be viewed as a relative version of the classical Noether inequality for surfaces. As a consequence, we give a new proof of the slope inequality for fibered surface without using any stability method. The treatment is essentially different from these of Xiao, Cornalba--Harris and Moriwaki. We also study the geography problem of surfaces in positive characteristics and show that the Severi inequality is true for surfaces of general type in positive characteristic whose Albanese map is generically finite. Moreover, the geography of surfaces with Albanese fibrations is studied.