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摘要翻译：
我们研究正则曲线模型上的群体作用。如果$x$是定义在完全D.V.R的分数域$k$上的光滑曲线。$R$,每个带有Galois群$G$的温和分支扩展$K'/K$在$X_{K'}$上诱导一个$G$-作用。本文研究了这种$G$-作用在某些正则的$X_{k'}$模型上的推广。特别地，我们得到了这种正则模型的特殊纤维结构束上同调群的交替和上的群元诱导的自同态的Brauer迹的公式。受这一全局研究的启发，我们还考虑了在一个驯服循环商奇点的例外轨迹上的Galois作用的类似问题。利用这些结果，我们用封闭的、幂次的子群格式，研究了特殊纤维的N\'eron Jacobian$x$N\'eron模型的自然过滤。我们证明了这种过滤中的跳变只依赖于严格法向交叉的最小正则模型的特殊纤维的纤维类型，特别是与剩余特性无关。在严格法向交叉为$x$超过$\spec(R)$时，这种跳变只依赖于最小正则模型的特殊纤维的纤维类型。此外，我们获得关于这些跳跃发生的位置的信息。对于属1和属2的曲线，我们还计算了有限多个可能的纤维类型中的每一种的跳跃。
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英文标题：
《Galois actions on models of curves》
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作者：
Lars Halvard Halle
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最新提交年份：
2007
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider similar questions for Galois actions on the exceptional locus of a tame cyclic quotient singularity. We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $\Spec(R)$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2.
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PDF链接：
https://arxiv.org/pdf/0711.1739
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