Fibred surfaces with general pencils of genus 5

TL;DR

This paper investigates fibred surfaces with genus 5 fibers, establishing a linear relation between key invariants involving the count of trigonal fibers, using the analysis of the relative canonical algebra.

Contribution

It introduces a new linear relation between invariants of fibred surfaces with genus 5 fibers, linking the canonical divisor square to the Euler characteristic and trigonal fibers.

Findings

Established the relation K_f^2 = χ_f + N for genus 5 fibred surfaces.

Connected the number of trigonal fibers to the surface's invariants.

Provided a method based on the analysis of the relative canonical algebra.

Abstract

Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=\chi_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\cal{R}(f)$.