LINEAR SYSTEMS ATTACHED TO CYCLIC INERTIA

MARCO A GARUTI

Abstract. We construct inductively an equivariant compactification of the algebraic group 𝕎_{n} of Witt vectors of finite length over a field of characteristic p > 0. We obtain smooth projective rational varieties \overline{𝕎}_{n}, defined over F_{p}; the boundary is a divisor whose reduced subscheme has normal crossings.

The Artin-Schreier-Witt isogeny F − 1 : 𝕎_{n} → 𝕎_{n} extends to a finite cyclic cover Ψ_{n} : \overline{𝕎}_{n} → \overline{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.