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

TL;DR

This paper investigates ramification invariants of Artin-Schreier extensions on smooth surfaces over algebraically closed fields of positive characteristic, providing a formula for computing a key ramification invariant and proving a conjecture on its bounds.

Contribution

It introduces a simple formula for computing the invariant r_x' associated with ramification and confirms Kato's conjecture on the upper bounds of r_x in this context.

Findings

Derived a formula to compute r_x' for Artin-Schreier extensions.

Proved Kato's conjecture on the upper bound of r_x.

Established equivalence of r_x' and r_x for good extensions.

Abstract

For a smooth surface X over an algebraically closed field of positive characteristic, we consider the ramification of an Artin-Schreier extension of X. A ramification at a point of codimension 1 of X is understood by the Swan conductor. A ramification at a closed point of X is understood by the invariant r_x defined by Kato [2]. The main theme of this paper is to give a simple formula to compute r_x' defined in [4], which is equal to r_x for good Artin-Schreier extension. We also prove Kato's conjecture for upper bound of r_x.