An Example About Semi-stable Fibration Of Genus 3 Over P~1

In this thesis,we mainly study the semi-stable fibration of surface.The proved Beauville's conjecture tells us if f:S-?P~1is a semi-stable fibration of curves of genus g>2 over P~1with s singular fibers,then s?5.By double cover,we construct a semi-stable fibration of curves of genus g=3 over P~1with 5 singular fibers,which reaches the lower bound for the number s of singular fibers.Moreover,we explicitly verify that the surface is a rational surface.