Recovering a covering of schemes from Galois orbits of power series

Paul Alexander Helminck
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)
2023-06-05 16:00:00
In this paper we study the problem of reconstructing the poset structure of a scheme using coverings. For a finite, dominant and separable morphism $X'\to{X}$ of normal, connected Noetherian schemes, we show that the poset structure of $X'$ can be recovered from the poset structure of $X$ and a set of Galois gluing data over $X$. This data consists of a set of local Galois orbits that are glued using relative Galois groups and transfer maps. To find this gluing data in practice, we introduce symbolic multivariate Newton-Puiseux algorithms that calculate adic approximations of the roots of a univariate polynomial over chains of prime ideals. By matching power series over intersecting chains of prime ideals, we are then able to glue the local Galois orbits of the approximations to obtain the global poset structure of the scheme. We show how to apply these algorithms to two types of coverings: coverings of semistable models of curves and sch\"{o}n coverings of very affine varieties. The first gives an algorithm to calculate the Berkovich skeleton of a curve, and the second an algorithm to calculate boundary complexes.
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