Linear systems attached to cyclic inertia. (English) Zbl 1072.14017

Garuti, Marco A.

Fried, Michael D. (ed.) et al., Arithmetic fundamental groups and noncommutative algebra. Proceedings of the 1999 von Neumann conference, Berkeley, CA, USA, August 16–27, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-2036-2/hbk). Proc. Symp. Pure Math. 70, 377-386 (2002).

Summary: 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 \(\overline{\mathbb{W}}_n\), defined over \(F_p\); the boundary is a divisor whose reduced subscheme has normal crossings.
The Artin-Schreier-Witt isogeny \(F-1: \mathbb{W}_n \rightarrow \mathbb{W}_n\) extends to a finite cyclic cover \(\Psi_n: \overline{\mathbb{W}}_n \rightarrow \overline{\mathbb{W}}_n\) 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 characteristic \(p\) at a wildly ramified point whose inertia group is cyclic.
For the entire collection see [Zbl 0993.00031].

MSC:

14E20	Coverings in algebraic geometry
11S15	Ramification and extension theory
13K05	Witt vectors and related rings (MSC2000)