The minimal number of singular fibers of a semistable curves over P^1

TL;DR

This paper proves Beauville's conjecture that non-trivial semistable fibrations of genus greater than one over P^1 have at least five singular fibers, and provides an explicit example for genus 2.

Contribution

It confirms a longstanding conjecture and constructs explicit examples, advancing understanding of the minimal singular fibers in semistable fibrations.

Findings

Proves Beauville's conjecture for genus g>1.

Constructs a genus 2 example with 5 singular fibers.

Establishes the minimal number of singular fibers as five.

Abstract

In this paper, we shall prove Beauville's conjecture: if $f:S \to P^1$ is a non-trivial semistable fibration of genus g>1, then $f$ admits at least 5 singular fibers. We have also constructed an example of genus 2 with 5 singular fibers. This paper will appear in the Journal of Algebraic Geometry.