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تسجيل مستخدم جديدConductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points
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Padmavathi Srinivasan
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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 $mathop{textrm{char}} k eq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = mathop{textrm{Spec}} R$. Let $mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $mathop{textrm{Art}} (mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $mathcal{X}$ and let $ u (Delta)$ denote the minimal discriminant of $C$. We prove that $-mathop{textrm{Art}} (mathcal{X}/S) leq u (Delta)$. As a corollary, we obtain that the number of components of the special fiber of $mathcal{X}$ is bounded above by $ u(Delta)+1$.
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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
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