Normalized Berkovich spaces and surface singularities

TL;DR

This paper introduces normalized Berkovich spaces over trivially valued fields, linking surface singularities to non-archimedean geometry, and characterizes valuations related to surface resolutions.

Contribution

It defines normalized Berkovich spaces as quotients by rescaling actions, and proves an analogue of Raynaud's theorem for these spaces, connecting formal schemes to non-archimedean models.

Findings

Normalized non-archimedean links of surface singularities resemble non-archimedean curves over $k((t))$

Characterization of essential and log essential valuations on surfaces

Establishment of a categorical equivalence for normalized Berkovich spaces

Abstract

We define normalized versions of Berkovich spaces over a trivially valued field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by rescaling semivaluations. We associate such a normalized space to any special formal $k$-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed $G$-topological space, which we prove to be $G$-locally isomorphic to a Berkovich space over the field $k((t))$ with a $t$-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of $k$-varieties, and allow to study the birational geometry of $k$-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field $k$ is analogous to the structure of non-archimedean analytic curves over $k((t))$, and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.