Linear systems attached to cyclic inertia

Abstract

We construct inductively an equivariant compactification of the algebraic group ${\mathbb W}_n$ of Witt vectors of finite length over a field of characteristic $p>0$. We obtain smooth projective rational varieties $\bar{\mathbb W}_n$, defined over $\mathbf F_p$; the boundary is a divisor whose reduced subscheme has normal crossings. The Artin-Schreier-Witt isogeny $F-1:{\mathbb W}_n\to {\mathbb W}_n$ extends to a finite cyclic cover ${\mathbfΨ}_n:\bar{\mathbb W}_n\to \bar{\mathbb W}_n$ of degree $p^n$ ramified at the boundary. This is used to give an extrinsic description of the local behavior of a separable cover of curves in char. $p$ at a wildly ramified point whose inertia group is cyclic. In an appendix, we give an elementary computation of the conductor of such a covering, which can otherwise be determined using class field theory.