The local lifting problem for actions of finite groups on curves

Details The local lifting problem for actions of finite groups on curves The local lifting problem for actions of finite groups on curves

2009-03-02

arxiv.org/abs/0903.0293v2

Mathematics

Algebraic Geometry

Scientific paper

Let $k$ be an algebraically closed field of characteristic $p > 0$. We study obstructions to lifting to characteristic 0 the faithful continuous action $\phi$ of a finite group $G$ on $k[[t]]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic 0 in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H \subset G$. We determine for which $G$ the KGB obstruction of every $\phi$ vanishes. We also consider analogous problems in which one requires only that an obstruction to lifting $\phi$ due to Bertin vanishes for some $\phi$, or for all sufficiently ramified $\phi$. These results provide evidence for a strengthening of Oort's lifting conjecture.

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