Moderately ramified actions in positive characteristic

Abstract

In characteristic 2 and dimension 2, wild $${\mathbb Z}/2{{\mathbb {Z}}}$$ -actions on k[[u, v]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic $$p>0$$ and dimension $$n\ge 2$$ arising from certain non-linear actions of $${\mathbb {Z}}/p{\mathbb {Z}}$$ on the formal power series ring $$k[[u_1,\dots ,u_n]]$$ . These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when $$p=2$$ . In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.