ON THE STABLE REDUCTION OF HYPERELLIPTIC CURVES

Let f : S → B Be a surface fiBration of genus g ≥ 2 over C. The semistaBle reduction theorem asserts there is a finite Base change π : B' → B such that the fiBration S ×B B' → B' admits a semistaBle model. An interesting invariant of f , denoted By N ( f ), is the minimum of deg( π) for all such π. In an early paper of Xiao, he gives a uniform multiplicative upper Bound Ng for N ( f ) depending only on the fiBre genus g. However, it is not known whether Xiao's Bound is sharp or not. In this paper, we give another uniform upper Bound N' g for N ( f ) when f is hyperelliptic. Our N' g is optimal in the sense that for every g ≥ 2 there is a hyperelliptic fiBration f of genus g so that N ( f ) = N' g . In particular, Xiao's upper Bound Ng is optimal when Ng = N' g. We show that this last equation Ng = N' g holds for infinitely many g.