Lifting Tropical Divisors with Prescribed Rank

Definitions and mathematical statement

In the standard interpretation in which $\Gamma$ is a compact metric graph, a divisor is a finite integral sum $D=\sum_{i=1}^{n}a_i p_i$, with degree $\deg D=\sum_{i=1}^{n}a_i=d$. A divisor is effective if all its coefficients are nonnegative. Linear equivalence is generated by divisors of continuous piecewise-linear functions with integral slopes. The divisor rank is $r_\Gamma(D)=-1$ if no effective divisor is linearly equivalent to $D$; otherwise, \[ r_\Gamma(D)=\max\left\{k\in{\mathbb{Z}}_{\geq0}: D-E\text{ is linearly equivalent to an effective divisor for every effective divisor }E\text{ of degree }k\right\}. \] These definitions specify the compact, unweighted metric-graph convention only; the supplied record does not establish whether that convention is intended, or whether vertex weights or unbounded edges are allowed.

For a divisor $\widetilde D$ on a smooth projective curve $X/K$, write \[ r_X(\widetilde D)=\dim_K H^0\left(X,\mathcal O_X(\widetilde D)\right)-1. \] A specialization map $\operatorname{sp}$ is understood to arise from a suitable model or skeleton of $X$, identified with the prescribed tropical curve. The exact model and identification are part of the unspecified meaning of "dual tropical curve" in the source.

Supplied lifting question. Given a tropical curve $\Gamma$ and a divisor \[ D=\sum_{i=1}^{n}a_i p_i,\qquad \deg D=d,\qquad r_\Gamma(D)=r, \] does there exist a non-Archimedean field $K$, a smooth projective curve $X/K$ whose dual tropical curve is $\Gamma$, and a divisor $\widetilde D$ on $X$, with multiplicity profile $(a_1,\ldots,a_n)$, such that \[ \operatorname{sp}(\widetilde D)=D \qquad\text{and}\qquad r_X(\widetilde D)=r? \] The intended quantification is over every input pair $(\Gamma,D)$ satisfying the stated conditions, with $K$, $X$, and $\widetilde D$ allowed to depend on that pair. This is a question, not an assertion that every pair admits a lift.

The source does not define "multiplicity profile." If the $p_i$ are distinct support points and the lifted support consists of distinct $K$-rational points $q_i$, its natural expression would be \[ \widetilde D=\sum_{i=1}^{n}a_i q_i,\qquad \operatorname{sp}(q_i)=p_i. \] This expression is conditional on that interpretation, not an additional supplied hypothesis. The record does not specify positivity of the $a_i$, rationality of lifted points, allowable field extensions, or treatment of residue degrees for closed points. Consequently, a fully determined lifting problem requires these conventions, together with the tropical-curve and specialization conventions, to be fixed.

Short English statement

The supplied question asks whether a tropical divisor can be lifted to a divisor on a smooth projective curve over some non-Archimedean field while preserving its specialization, multiplicity profile, and rank exactly. The meanings of tropical curve, dual tropical curve, and multiplicity profile require clarification before the question has a single precise interpretation.

Sources

• Local UnsolvedMath registry: permanent ID $30003965$, title Lifting Tropical Divisors with Prescribed Rank, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The supplied record gives proposed year $2018$ and names no individual proposer.
• Supplied source citation: Flat Surfaces and Algebraic Curves (2019), Oberwolfach Reports, pp. 2583--2650, DOI 10.4171/owr/2018/42. Supplied source URL and literature-source link labelled "OWR report": https://doi.org/10.4171/owr/2018/42. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Tropical canonical divisors, https://arxiv.org/abs/1710.06401.
• Supplied literature source: Tropical pluri-canonical divisors, https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/jlms.70027.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, describes the general realizability problem as difficult by "Murphy law," states that canonical and pluri-canonical cases have criteria, and states that no general solution is known. This is supplied dated curation, not an independently verified literature assessment; the supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The supplied registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review. These are source metadata, not independent verification of the mathematical formulation or its status.