Ramification of surfaces: sufficient jet order for wild jumps

TL;DR

This paper introduces a new method to determine the minimal jet order needed to capture wild ramification jumps in algebraic surface coverings, linking geometric tangency to ramification invariants.

Contribution

It shows that wild ramification jumps depend only on a finite jet of the curve, with bounds related to tangency order, advancing understanding of ramification in positive characteristic surfaces.

Findings

Ramification jumps depend only on the jet of the curve.

Upper bounds for jet order are linearly related to tangency.

Results apply to abelian p-groups and contribute to Deligne's program.

Abstract

We develop a new approach to construction of numerical invariants for ramified coverings of algebraic surfaces of prime characteristic. Let A be a two-dimensional regular local ring of prime characteristic p with algebraically closed residue field. Let L/K be a solvable finite Galois extension of its fraction field. Let C be a germ of a regular curve on the spectrum of A which is not a component of the ramification divisor R. Then L/K determines (up to conjugation) an extension of the function filed of C. This is an extension of discrete valuation fields with algebaically closed residue fields, and we consider ramification jumps of this extension. Varying C, we obtain a collection of data describing the ramification of L/K with respect to C. In the present paper we show that the above mentioned ramification jumps depend only on the jet of C of certain order, and the upper bound for this order depends linearly on the order of tangence of C and R. Strictly speaking, this is done for wild ramification jumps, i.e., for the ramification jumps divided by the order of tame ramification. It would be important to prove a stronger fact: not only the jumps but the whole ramification filtration is the same when C is running over a jet of certain order. This statement (for curves transversal to R) is a step in Deligne's program describing how to compute Euler-Poincare characteristics of constructible etale sheaves on surfaces. In the present version we prove the above fact for some class of Galois groups that includes, in particular, all abelian p-groups.