On the Monodromy of Rational Singularities in Mixed Characteristic

We study the ramification on the cohomology of a smooth proper surface $X$ in mixed characteristic, when $X$ degenerates to a surface over $\overline{\mathbb{F}}_p$ with rational singularities, with a focus on the case of rational double points. We find that the associated monodromy action of inertia depends on a formal affine neighborhood of the singularity, and under sufficient restrictions on characteristic $p$, it is tamely ramified and generated by a conjugacy class representative of an appropriate Weyl group related to the singularity. This naturally extends to a similar monodromy characterization of general rational singularities. Along the way we extend to mixed characteristic some results of Brieskorn and Slodowy concerning simultaneous resolutions of surface singularities. We also compare our Weyl group actions to certain Springer representations constructed by Borho and MacPherson, via the notion of relative perversity as developed by Hansen and Scholze.