b-divisorial valuations and Berkovich positivity functions
Joaquim Roé, Stefano Urbinati
TL;DR
The paper develops a unifying framework linking Berkovich analytification, the Zariski–Riemann space, and Shokurov b-divisors to study positivity invariants of divisors as one varies over valuation spaces. It defines a canonical b-divisor Dξ attached to each seminorm ξ, proves that Dξ is lower semicontinuous and that Dξ varies continuously on quasimonomial (Abhyankar) families, and extends Seshadri constants and asymptotic orders of vanishing to seminorms. The results connect valuation-theoretic data with nef envelopes and asymptotic multiplier ideals, establishing a robust translation from valuative to positivity information and highlighting special structure in the surface case where explicit formulas for Dξ are obtained. Collectively, the work provides a cohesive approach to understanding Nagata-type and Newton–Okounkov-type discontinuities through the lens of b-divisorial valuations and their positivity cones.
Abstract
We prove semicontinuity properties for local positivity invariants of big and nef divisors. The usual definition of Seshadri constant and asymptotic order of vanishing along a subvariety is extended to include all seminorms in the Berkovich space, and we obtain semicontinuity of such constants as a function of the center seminorm. We use Shokurov's language of b-divisors; to each seminorm there is an associated b-divisor which can be used to translate questions about positivity into questions about the shape of certain cones of b-divisors. The theory works especially well for what we call b-divisorial valuations, a natural extension of the notion of divisorial valuations which encompasses, e.g., all Abhyankar valuations.