Conductors and minimal discriminants of hyperelliptic curves: A comparison in the tame case

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 $\mathrm{char}k>2g+1$. Let $S=\mathrm{Spec}R$. Let $X$ be the minimal proper regular model of $C$ over $S$. Let $\mathrm{Art}(C/K)$ denote the Artin conductor of the $S$-scheme $X$ and let $\nu ({\Delta }_C)$ denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}(C/K)\le \nu ({\Delta }_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent refinement of Abhyankar's inversion formula for studying plane curve singularities. We also prove combinatorial restrictions for $-\mathrm{Art}(C/K)=\nu ({\Delta }_C)$.

References in corpus (2)

• Conductor and discriminant of Picard curves
• Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points