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arXiv · 2511.22600

b-divisorial valuations and Berkovich positivity functions

Joaquim Roé·Stefano Urbinati
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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.

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Joaquim Roé, Stefano Urbinati. 2025-11-27. b-divisorial valuations and Berkovich positivity functions. https://arxiv.org/abs/2511.22600

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