Polyhedral Global Newton–Okounkov Bodies for Mori Dream Spaces

Definitions and mathematical statement

Let $X$ be a smooth Mori dream space of dimension $n$. In the usual projective convention, this means a smooth projective variety for which $\operatorname{Pic}(X)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}\cong N^1(X)_{{\mathbb{Q}}}$ and the Cox ring, the algebra of global sections graded by divisor classes, is finitely generated. Here $N^1(X)_{{\mathbb{R}}}$ denotes the real vector space of Cartier divisors modulo numerical equivalence, and $N^1(X)_{{\mathbb{Q}}}$ its rational structure. The source does not specify the ground field or its characteristic; the intended field hypotheses remain unresolved.

An admissible flag is a chain of irreducible closed subvarieties \[ Y_\bullet:\quad X=Y_0\supset Y_1\supset\cdots\supset Y_n=\{x\}, \qquad \operatorname{codim}_X(Y_i)=i, \] with every $Y_i$ nonsingular at $x$. For a Cartier divisor $D$ and a nonzero section $s\in H^0(X,\mathcal O_X(D))$, let $\nu_{Y_\bullet}(s)\in{\mathbb{Z}}_{\geq 0}^{n}$ be its flag valuation: successively take the order of vanishing along the next member of the flag, divide out that vanishing, and restrict to that member.

Using the standard closed-cone convention, define the global Newton--Okounkov body by \[ \Delta_{Y_\bullet}(X) = \overline{\operatorname{cone}\!\left\{ \left(\nu_{Y_\bullet}(s),[D]\right): D\text{ a Cartier divisor on }X,\quad 0\ne s\in H^0(X,\mathcal O_X(D)) \right\}} \subseteq {\mathbb{R}}^n\times N^1(X)_{{\mathbb{R}}}. \] Here $\operatorname{cone}$ means finite nonnegative real linear combinations, and the bar denotes closure. This global object is a cone, despite the customary name "body." It is rational polyhedral if there are finitely many vectors $v_1,\ldots,v_r\in{\mathbb{Q}}^n\times N^1(X)_{{\mathbb{Q}}}$ such that \[ \Delta_{Y_\bullet}(X)=\sum_{j=1}^{r}{\mathbb{R}}_{\geq0}v_j. \]

Question. Does every smooth Mori dream space $X$ admit an admissible flag $Y_\bullet$ for which $\Delta_{Y_\bullet}(X)$ is rational polyhedral?

The quantifier is over smooth Mori dream spaces, as in both supplied problem formulations. The background assessment instead discusses all smooth projective varieties; that broader class is not substituted into the question. The ground-field hypotheses, including any intended rationality requirement on the terminal flag point, require source clarification.

Short English statement

Does every smooth Mori dream space have an admissible flag making its global Newton--Okounkov cone rational polyhedral? The supplied record labels this question open. Its ground field is unspecified, and its broader background assessment does not establish the status of this precise question.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003596$, source problem code catalogue entry, titled "Polyhedral Global Newton--Okounkov Bodies for Mori Dream Spaces." No individual proposer is supplied; the recorded proposed year is 2017.
• Supplied report citation: Mini-Workshop: Positivity in Higher-dimensional Geometry: Higher-codimensional Cycles and Newton--Okounkov Bodies (2018), Oberwolfach Reports, pp. 2631--2657, DOI 10.4171/owr/2017/43. Supplied source link: https://doi.org/10.4171/owr/2017/43. The citation's 2018 date and the record's 2017 year have not been reconciled.
• Supplied literature reference: Schmitz--Seppanen, "global Okounkov bodies," https://doi.org/10.1016/j.jalgebra.2017.07.018.
• Supplied literature reference: Postinghel--Urbinati, "Newton--Okounkov bodies," https://arxiv.org/abs/1306.2028.
• Supplied literature reference: "original OWR report," https://doi.org/10.4171/OWR/2017/43.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and reports rational polyhedral global Newton--Okounkov bodies for toric varieties, Mori dream spaces with suitable flags, and several homogeneous families, while reporting no theorem providing such a flag for every smooth projective variety. The supplied background describes this as dated literature triage derived from OWR curation and requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with the explanation that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is registry metadata, not independent mathematical or source verification by this draft.