b-divisorial valuations and Berkovich positivity functions

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.

💡 Research Summary

This paper establishes a profound connection between the Berkovich analytification of a variety, the Riemann-Zariski space, and the theory of b-divisors in birational geometry. The primary goal is to study the semicontinuity properties of local positivity invariants, specifically Seshadri constants and asymptotic orders of vanishing, for big and nef divisors. The authors achieve this by extending the classical definitions of these invariants from subvarieties to all seminorms in the Berkovich space of a smooth projective variety X.

The core construction is the association of a b-divisor D_ξ to any seminorm ξ in the Berkovich space X^an. This D_ξ is defined as the limit of b-divisors associated to the graded sequence of valuation ideals defined by ξ. This translation allows questions about positivity measured by ξ to be reformulated in terms of the geometry of cones of b-divisors. The theory is particularly effective for a class of valuations the authors term “b-divisorial.” These are valuations for which the associated b-divisor D_ξ is non-zero, allowing the recovery of the original valuation from this limiting divisor. This class strictly contains quasimonomial (Abhyankar) valuations and hence all divisorial valuations. A key characterization (Theorem 1.1) shows that in dimension two, a valuation is b-divisorial if and only if D_ξ ≠ 0, and in any dimension, valuations with sublinear log-discrepancy are b-divisorial.

The paper’s first major analytical result (Theorem 1.2) concerns the map ξ → D_ξ. It is proven that this map from the Berkovich space to the space of b-divisors is lower semicontinuous. Furthermore, its restriction to the interior of any simplex Δ of quasimonomial valuations (defined with respect to a local system of parameters on a model) is continuous. This provides a crucial link between the topology of the valuation space and the geometry of b-divisors.

Using this foundation, the authors define the asymptotic order of vanishing ω(D, ξ) and the Seshadri constant ε(D, ξ) for an ample divisor D and a b-divisorial valuation v_ξ via the associated b-divisor: ω(D, ξ) = sup{t | D + tD_ξ is pseudoeffective} and ε(D, ξ) = sup{t | D + tD_ξ is nef}. The main continuity results (Theorem 1.3) are then established: the function ξ → ω(D, ξ) is lower semicontinuous on the Berkovich space and continuous on the interior of any quasimonomial simplex Δ. The function ξ → ε(D, ξ) is continuous on the interior of any such Δ. While ε is not globally semicontinuous in general, a special result is proven for surfaces: if X is a surface and p is a smooth point, then ε is lower semicontinuous over the subspace X_p of valuations centered at p.

The work is motivated by pathological discontinuity phenomena observed in seemingly disparate contexts: the study of Nagata’s conjecture on plane curves, symplectic ellipsoid packing, and the jumping of Newton-Okounkov bodies. By framing these problems within the unified language of b-divisors associated to Berkovich seminorms, the paper offers a geometric explanation for these discontinuities. It demonstrates that the behavior of positivity invariants is intrinsically linked to the structure of the valuation space and the associated b-divisors. The introduction of b-divisorial valuations provides a natural setting to unify the analysis, especially for non-flag valuations on surfaces, paving the way for future investigations into the continuity properties of Newton-Okounkov bodies and deeper connections with symplectic geometry.