The minimal number of singular fibers of a semistable curves over P¹

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.