An Arakelov theoretic proof of the equality of conductor and discriminant

On Szpiro's Discriminant Conjecture

We consider generalizations of Szpiro's classical discriminant conjecture to hyperelliptic curves over a number field $K$, and to smooth, projective and geometrically connected curves $X$ over $K$ of genus at least one. The main results give effective exponential versions of the generalized conjectures for some curves, including all curves $X$ of genus one or two. We obtain in particular exponential versions of Szpiro's classical discriminant conjecture for elliptic curves over $K$. In course of our proofs we establish explicit results for certain Arakelov invariants of hyperelliptic curves (e…

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

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 $ν(Δ_C)$ denote the minimal discriminant of $C$. We prove that $-\mathrm{Art}\ (C/K) \leq ν(Δ_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent…

Conductors and minimal discriminants of hyperelliptic curves with rational Weierstrass points

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 $\mathop{\textrm{char}} k \neq 2$. 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 $ν(Δ)$ denote the minimal discriminant of $C$. We prove that $-\mathop{\textrm{Art}} (\mathcal{X}/S) \leq ν(Δ)$. As a …

Strict Arakelov inequality for a family of varieties of general type

Let $f:\, X\to Y$ be a semistable non-isotrivial family of $n$-folds over a smooth projective curve with discriminant locus $S \subseteq Y$ and with general fibre $F$ of general type. We show the strict Arakelov inequality \[\frac{\mathrm{deg}\, f_*ω_{X/Y}^ν}{\mathrm{rank}\, f_*ω_{X/Y}^ν} < {nν\over 2}\cdot\mathrm{deg}\,Ω^1_Y(\log S),\] for all $ν\in \mathbb N$ such that the $ν$-th pluricanonical linear system $|ω^ν_F|$ is birational. This answers a question asked by Möller, Viehweg and the third named author.

奇剩余特征下超椭圆曲线的导子-判别式不等式

Conductor-discriminant inequality for hyperelliptic curves in odd residue characteristic

本文证明了在特征不为2的奇剩余特征下，离散赋值域上超椭圆曲线的导子与判别式之间的不等式，将Artin导子的负值与特殊纤维不可约分支数量由判别式赋值上界控制，推广了前人工作。

导子-判别式不等式I：缓分歧循环覆盖

Conductor-Discriminant Inequality for Tamely Ramified Cyclic Covers I

本文证明了离散赋值域上Z/n-覆盖的P^1的导子-判别式不等式，推广了超椭圆曲线情形，并加强了Kohls关于Jacobian导子指数的结果。