ON THE STABLE REDUCTION OF HYPERELLIPTIC CURVES

作者：Gong, Cheng; Gu, Yi^*; Lu, Jun; Pollack, Paul

来源：Tohoku Mathematical Journal, 2022, 74(2): 195-213.

DOI：10.2748/tmj.20201126

• Hyperelliptic surface fibration
• semistable reduction
• monodromy

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 pi : B' -> B such that the fibration S x(B) B' -> B' admits a semistable model. An interesting invariant of f , denoted by N (f), is the minimum of deg (pi) for all such pi. In an early paper of Xiao, he gives a uniform multiplicative upper bound N-g 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 N-g is optimal when N-g = N-g'. We show that this last equation N-g = N-g' holds for infinitely many g.

• 单位
  y; 1; 苏州大学