曲线上有限可解Torsons的扩张
Extention of Finite Solvable Torsors over a Curve
摘要：
设$R$是具有分式域$K$和正特征$p$的代数闭剩余域的离散赋值环.设$X$是$R$上的光滑纤维曲面，其上的纤维几何连接，并具有截面$x\in X(R)$。设$G$是一个有限可解的$K$ -群概型，并假定$|G|=p^n$或$G$都有长度为2的正规级数.证明了$X$的一般纤维$X_η$上的每一个商点$G$ -torsor在最终扩展标量和最终在其特殊纤维$X_s$的闭子概型上爆破$X$之后都可以扩展为$X$上的torsor.
英文摘要：
Let $R$ be a discrete valuation ring with fraction field $K$ and with algebraically closed residue field of positive characteristic $p$. Let $X$ be a smooth fibered surface over $R$ with geometrically connected fibers endowed with a section $x\in X(R)$. Let $G$ be a finite solvable $K$-group scheme and assume that either $|G|=p^n$ or $G$ has a normal series of length 2. We prove that every quotient pointed $G$-torsor over the generic fiber $X_η$ of $X$ can be extended to a torsor over $X$ after eventually extending scalars and after eventually blowing up $X$ at a closed subscheme of its special fiber $X_s$.
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