2022 On the stable reduction of hyperelliptic curves
Cheng Gong, Yi Gu, Jun Lu, Paul Pollack
Tohoku Math. J. (2) 74(2): 195-213 (2022). DOI: 10.2748/tmj.20201126


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 $g$.


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Cheng Gong. Yi Gu. Jun Lu. Paul Pollack. "On the stable reduction of hyperelliptic curves." Tohoku Math. J. (2) 74 (2) 195 - 213, 2022. https://doi.org/10.2748/tmj.20201126


Published: 2022
First available in Project Euclid: 6 July 2022

Digital Object Identifier: 10.2748/tmj.20201126

Primary: 14D05
Secondary: 11J71 , 14D06 , 14J25

Keywords: Hyperelliptic surface fibration , Monodromy , semistable reduction

Rights: Copyright © 2022 Tohoku University


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Vol.74 • No. 2 • 2022
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