On ramifications of Artin-Schreier extensions of surfaces over algebraically closed fields of positive characteristic I

Abstract

We give an algorithm which computes $r$, defined by K. Kato in the paper [1], which is an important invariant for Artin-Schreier extensions of surfaces $X$ over fields of positive characteristic. The Swan conductor gives the invariant of ramifications concerning codimension 1 subvarieties of $X$. This $r$ gives the invariant of ramifications concerning codimension 2 subvarieties of $X$. The invariant $r$ is important to calculate the Euler Poincaré characteristic of some smooth $l$-adic sheaf of rank 1 on an open dense subscheme $U$ of $X$.