Recovering the combinatorial structure of a covering of schemes using glued Galois categories

In this talk, I will show how one can reconstruct the relative poset structure of a finite separable covering of a fixed normal connected Noetherian scheme in terms of glued Galois categories. This relies on a local transitivity result for the Galois action on chains of length one, which generally does not hold for longer chains. The 2-categorical gluing data connecting the various local Galois categories leads to two distinct types of monodromy, which I will illustrate with several explicit examples.

As two applications of this technique, I will show how it can be used to
1. Find the p-adic semistable reduction type of a modular curve.
2. Obtain a full algorithm to compute dual intersection graphs of semistable models of curves.

Determining the monodromy in each of these cases forms an important step toward obtaining the final global combinatorial structure, as different monodromy types in general lead to different graphs.