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 nonarchimedean models for the links of the singularities of $k$varieties, and allow to study the birational geometry of $k$varieties using techniques of nonarchimedean geometry available only when working over a field with nontrivial valuation. In particular, we prove that the structure of the normalized nonarchimedean links of surface singularities over an algebraically closed field $k$ is analogous to the structure of nonarchimedean 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2014
 DOI:
 10.48550/arXiv.1412.4676
 arXiv:
 arXiv:1412.4676
 Bibcode:
 2014arXiv1412.4676F
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 Several corrections and improvements. New result added, Theorem 10.8, proving a characterization of essential valuations. 51 pages. To appear in Transactions of the AMS