Recovering the relative poset structure of a covering of schemes using glued topoi

Abstract: Let $X$ be a normal connected Noetherian scheme. In this paper we give an algorithm to reconstruct the relative poset structure of finite dominant separable morphism $X'\to X$ in terms of topos-theoretic enhancements of the underlying poset of $X$. The different relative poset structures are classified by an object in a $2$-limit of topoi over a graph, which we call the universal poset covering. We interpret the data in the universal covering in terms of glued power series, and we show how to calculate these using iterated symbolic multivariate Newton-Puiseux algorithms. This in particular gives a full local monogenic algorithm to calculate dual intersection graphs of semistable models of curves defined over a discretely valued field. We give a detailed study of this algorithm, with various examples to illustrate the non-trivial gluing phenomena. We also interpret these techniques in terms of analytic spaces, with an eye towards future applications in $p$-adic integration theory.