Trigonal Canonically Fibered Surfaces

TL;DR

This paper refines the proof of Xiao's conjecture for canonically fibered surfaces of genus 5, simplifies the argument, and establishes new inequalities for surfaces of genus 3 and 4.

Contribution

It provides a simplified proof of Xiao's conjecture for genus 5 surfaces and extends the approach to derive inequalities for genus 3 and 4 surfaces.

Findings

Improved bounds on the geometric genus of fibered surfaces

Simplified proof of Xiao's conjecture for genus 5

New Noether-type inequalities for genus 3 and 4 surfaces

Abstract

We fix some gaps of a proof of Xiao's conjecture on canonically fibered surfaces of relative genus 5 by the second author. Our argument simplifies the original proof and gives a much better bound on the geometric genus of the surface. Also we apply the same argument to canonically fibered surfaces of relative genus 3 and 4 to obtain some Noether-type inequalities for these surfaces.