Normalized Berkovich spaces and surface singularities

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.