b-divisorial valuations and Berkovich positivity functions

2025/11/27 by Joaquím Roé, Roé, Joaquim, Stefano Urbinati +1
Mathematics · #Advanced Topology and Set Theory #Advanced Banach Space Theory #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2511.22600

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.

Citations

• Singular algebraic curves and infinite symplectic staircases
• Convexity of multiplicities of filtrations on local rings
• A non-Archimedean approach to K-stability, I: Metric geometry of spaces of test configurations and valuations
• Newton-Okounkov bodies sprouting on the valuative tree
• Valuations and asymptotic invariants for sequences of ideals
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• Singularities of Pairs

Related

• Seshadri constants and Okounkov bodies revisited
• Seshadri constants for curve classes
• Extended Okounkov bodies and multi-point Seshadri constants
• Bounds for Seshadri Constants
• Positive cones of b-divisor classes