Ramification and sufficient jet orders

Our work is devoted to ramification theory of n-dimensional schemes with n ≥ 2. We start to develop an approach which is very natural; however, it has not got enough attention, except for [Del] and [Bry83]. The idea is to reduce the situation of an n-dimensional scheme X to a number of 1-dimensional settings, just by restricting to curves C properly crossing the ramification subscheme R ⊂ X . What is new here is that we do not require C to be transversal to R. Consideration of curves tangent to the ramification divisor seems to yield more important information about the “higher-dimensional ramification”. In the current paper we restrict the setting as follows: • n = 2; • X is equidimensional, i. e., it can be equipped with a morphism to Spec k, where k is an algebraically closed field of prime characteristic p; • If P is any closed point of X , the local ring OX ,P is a 2-dimensional excellent local ring with the residue field k; • X is regular. In section 2 we state a series of questions related to the behavior of ramification jumps as one varies the curve C. Some of these questions are new and some generalize the questions considered in the above mentioned papers. In the following sections we answer the first of these questions affirmatively. Namely, we prove that the ramification invariants depend only on the jet of C of certain order, and we can bound this order in a certain uniform way. This statement is a step in Deligne’s program [Del] describing how to compute EulerPoincaré characteristics of constructible étale sheaves on surfaces. As for the other questions, in the case of Artin-Schreier extensions we can answer affirmatively to the most of them. This will be the subject of another paper [Zhu].