Realizability of Divisor Volumes

Definitions and mathematical statement

Let $X$ be a projective variety over a field $K$, and write $n=\dim X$. Under the interpretation that an integral divisor means a Cartier divisor with integer coefficients, define \[ h^0(X,mD):=\dim_K H^0\left(X,\mathcal O_X(mD)\right), \qquad \operatorname{vol}_X(D):= \limsup_{\substack{m\to\infty\\m\in{\mathbb{Z}}_{>0}}} \frac{n!\,h^0(X,mD)}{m^n}. \] Here $\mathcal O_X(mD)$ is the invertible sheaf associated with the Cartier divisor $mD$.

The supplied realizability question is: does every $r\in{\mathbb{R}}_{\geq 0}$ admit a projective variety $X$ and an integral divisor $D$ on $X$ such that \[ \operatorname{vol}_X(D)=r? \] Both $X$ and $D$ may depend on $r$; the source fixes no dimension and imposes no explicit smoothness, normality, ampleness, or bigness hypothesis.

The supplied record does not specify a ground field, whether that field is fixed throughout the question, or whether "integral divisor" means an integral Cartier divisor or an integral Weil divisor. Consequently, the displayed definition gives the Cartier interpretation, while the admissible class of varieties and divisors remains unresolved. A Weil-divisor interpretation would require additional hypotheses or conventions for the spaces of sections. These choices must be checked against the source before this becomes a fully specified mathematical statement.

Short English statement

The question asks whether every nonnegative real number can be realized as the asymptotic volume of an integral divisor on a projective variety, allowing the variety and its dimension to vary. The ground field and the precise meaning of integral divisor are unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001870$, title "Realizability of Divisor Volumes," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2011$ and supplies no named proposer. It labels the statement "exact" and reports a comparison with the official report; that verification is supplied provenance, not an independent verification here.
• Supplied source citation, also listed as "original OWR report": Mini-Workshop: New Developments in Newton-Okounkov Bodies (2012), Oberwolfach Reports, pp. 2327--2363, DOI 10.4171/owr/2011/41. https://doi.org/10.4171/owr/2011/41. The background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source: Convex bodies associated to linear series. https://arxiv.org/abs/0805.3885.
• Supplied literature source: asymptotic invariants of line bundles. https://arxiv.org/abs/math/0505054.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved." It states that Riemann--Roch and asymptotic volume theory provide substantial asymptotic information, but that no uniform formula for $h^0(X,kD)$ on arbitrary projective varieties with Cartier divisors is known. This is supplied dated curation, not independently verified; the record explicitly requests independent verification of time-sensitive status and cited claims. Its relevance to the precise volume-realizability question requires review.