论文标题
扩展扭转曲线的弯曲
Extending torsors over regular models of curves
论文作者
论文摘要
让$ r $成为一个离散的评估戒指，其片段$ k $和残留的字段$ k $的特征$ p> 0 $。鉴于有限的交换集团方案$ g $超过$ k $，并且具有理性点的光滑投影曲线$ c $ cub over $ k $，因此我们研究了尖头的FPPF $ g $ -torsors的扩展，而不是$ c $ to $ c $ to pointed torsors to tosed torsors，而不是一些$ r $ r $ r $的型号$ \ mathcal $ \ mathcal {c} $ c $ $ c $。我们首先在日志方案类别中研究了此问题：给定有限的平面$ r $ - 组方案$ \ Mathcal {g} $，我们证明了一个指向$ \ Mathcal {g} $ - log-log Torsor的数据超过了$ \ Mathcal {C} $，与Morphism $ \ Mathcal c}相当于\ Mathrm {pic}^{log} _ {\ Mathcal {c}/r} $，其中$ \ Mathcal {g}^d $是$ \ Mathcal {G} $的Cartier dual，以及$ \ Mathcal {g} $和$ \ \ \ \ \ \\ Mathrm {pic}^log}^{log}^{log} _}然后，我们推断出扩展Torsors的标准：找到一个有限的平面型号，$ g $超过$ r $，对于Jacobian $ j $ $ c $的某种群体方案形态扩展到$ J $的Néron型号。在这种情况下，我们计算出来自FPPF的延伸日志Torsor的阻塞。在第二部分中，我们概括了Chiodo的结果，该结果给出了$ j $的Néron模型的$ r $ torsion子组，以成为有限的扁平组计划，我们将其与第一部分的结果结合在一起。最后，当$ c $是$ \ mathbb {q} $定义的过度椭圆形曲线时，我们给出了两个详细的torsors扩展的示例，这将说明我们的技术。
Let $R$ be a discrete valuation ring with field of fractions $K$ and residue field $k$ of characteristic $p>0$. Given a finite commutative group scheme $G$ over $K$ and a smooth projective curve $C$ over $K$ with a rational point, we study the extension of pointed fppf $G$-torsors over $C$ to pointed torsors over some $R$-regular model $\mathcal{C}$ of $C$. We first study this problem in the category of log schemes: given a finite flat $R$-group scheme $\mathcal{G}$, we prove that the data of a pointed $\mathcal{G}$-log torsor over $\mathcal{C}$ is equivalent to that of a morphism $\mathcal{G}^D \to \mathrm{Pic}^{log}_{\mathcal{C}/R}$, where $\mathcal{G}^D$ is the Cartier dual of $\mathcal{G}$ and $\mathrm{Pic}^{log}_{\mathcal{C}/R}$ the log Picard functor. Then, we deduce a criterion for the extension of torsors: it suffices to find a finite flat model of $G$ over $R$ for which a certain group scheme morphism to the Jacobian $J$ of $C$ extends to the Néron model of $J$. In this context, we compute the obstruction for the extended log torsor to come from an fppf one. In a second part, we generalize a result of Chiodo which gives a criterion for the $r$-torsion subgroup of the Néron model of $J$ to be a finite flat group scheme, and we combine it with the results of the first part. Finally, we give two detailed examples of extension of torsors when $C$ is a hyperelliptic curve defined over $\mathbb{Q}$, which will illustrates our techniques.
