Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points

Abstract

Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $\mathrm{char}k\ne 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S=\mathrm{Spec}R$. Let $\mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $\mathrm{Art}(\mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $\mathcal{X}$ and let $\nu (\Delta )$ denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}(\mathcal{X}/S)\le \nu (\Delta )$. As a corollary, we obtain that the number of components of the special fiber of $\mathcal{X}$ is bounded above by $\nu (\Delta )+1$.

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