Tohoku Mathematical Journal
2022
June
SECOND SERIES VOL. 74, NO. 2

Tohoku Math. J.
74 (2022), 195-213

Title ON THE STABLE REDUCTION OF HYPERELLIPTIC CURVES

Author Cheng Gong, Yi Gu, Jun Lu and Paul Pollack

(Received January 22, 2020, revised November 24, 2020)
Abstract. Let $f: S\to B$ be a surface fibration of genus $g\ge 2$ over ${\mathbb{C}}$. The semistable reduction theorem asserts there is a finite base change $\pi: B'\to B$ such that the fibration $S\times_BB'\to B'$ 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 $g\ge 2$ 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 $g$.

Mathematics Subject Classification. Primary 14D05; Secondary 14D06, 14J25, 11J71.
Key words and phrases. Hyperelliptic surface fibration, semistable reduction, monodromy.

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