Three point covers with bad reduction

We study Galois covers of the projective line branched at three points with bad reduction to characteristic p, under the condition that p exactly divides the order of the Galois group. As an application of our results, we prove that the field of moduli of such a cover is at most tamely ramified at p.Comment: 37 page

Formal deformation of curves with group scheme action

We study equivariant deformations of singular curves with an action of a finite flat group scheme, using a simplified version of Illusie's equivariant cotangent complex. We apply these methods in a special case which is relevant for the study of the stable reduction of three point covers.Comment: 44 pages, revised versio

Wild ramification kinks

Given a branched cover $f:Y\to X$ between smooth projective curves over a non-archimedian mixed-characteristic local field and an open rigid disk $D\subset X$, we study the question under which conditions the inverse image $f^{-1}(D)$ is again an open disk. More generally, if the cover $f$ varies in an analytic family, is this true at least for some member of the family? Our main result gives a criterion for this to happen.Comment: Final version, to appear in Research in the Mathematical Sciences. 29 page