Tropical Curve Counts for Smooth Cubic Log Divisors

Definitions and mathematical statement

Let $E\subset{\mathbb{P}}^2$ be a smooth plane cubic, and let $X,M_E$ denote $X={\mathbb{P}}^2$ with the divisorial logarithmic structure determined by $E$: its local sections are regular functions that are invertible away from $E$. This logarithmic pair is denoted by ${\mathbb{P}}^2(\log E)$ in the supplied record. The base field is not specified in that record.

Logarithmic Gromov--Witten invariants of this pair are virtual curve invariants defined using moduli spaces of stable logarithmic maps to $(X,M_E)$, together with specified discrete data and insertions. Such data include the genus, curve class, marked points, and contact orders along $E$. Denote a chosen invariant, once its conventions and admissible data $\mathcal D$ have been fixed, by $\operatorname{GW}^{\log}_{\mathcal D}({\mathbb{P}}^2,E)$.

Construction and correspondence problem. For a smooth cubic plane curve $E$, construct an appropriate tropical space $\mathcal T_E$, a notion of tropical curves in that space, and a counting rule with specified multiplicities and matching constraints, such that \[ N^{\mathrm{trop}}_{\mathcal D}(\mathcal T_E) = \operatorname{GW}^{\log}_{\mathcal D}({\mathbb{P}}^2,E) \] for the intended range of corresponding enumerative data $\mathcal D$.

Unresolved scope. The supplied statement does not specify the structure required of $\mathcal T_E$, the tropical counting rule, the base field, the range of genera and curve classes, the contact conditions, the insertions, or the conventions for the logarithmic invariants. Thus the displayed equality records the requested correspondence, rather than a fully quantified conjecture. These choices and their admissibility conditions must be supplied before a precise theorem can be stated.

The original excerpt motivates changing the logarithmic divisor through a toric correspondence slogan: tropical curves in ${\mathbb{R}}^n$ and algebraic curves in a projective toric variety have matching nonnegative counts when constraints, genus, and degree match. It supplies no hypotheses making that slogan a general theorem. In particular, it does not establish nonnegativity, integrality, or an unweighted enumerative interpretation for the logarithmic invariants sought here.

Short English statement

Construct a tropical space and a curve-counting theory corresponding to logarithmic Gromov--Witten invariants of the projective plane with a smooth cubic boundary. The source leaves the precise tropical structure, invariant conventions, and range of curve data unspecified.

Sources

• Local registry: supplied UnsolvedMath record in data/unsolvedmath/problems.json, permanent ID $30005536$, exact source problem code catalogue entry, titled Tropical Curve Counts for Smooth Cubic Log Divisors. The record belongs to the Oberwolfach Reports Open Problems collection and supplies proposed year $2023$; no individual proposer is identified.
• Supplied report citation: Tropical Methods in Geometry (2023), Oberwolfach Reports, DOI 10.4171/owr/2023/23, pp. 1263--1322. https://doi.org/10.4171/owr/2023/23.
• Supplied literature source, labeled OWR 2023 23: https://ems.press/content/serial-article-files/47020.
• The original excerpt attributes its toric correspondence slogan to the reference labels B17, CFPU16, CJMR17, GM07, GPS10, G10, G18, GS22, IKS03, LR18, M05, M07, MR09, MR20, NS06, O15, and R17, "and more." Bibliographic details for these labels are not supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that changing from the toric boundary to a smooth cubic changes the logarithmic space, with the requested general tropical correspondence remaining unproved. This is supplied dated curation, not an independently verified assessment.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. These are supplied verification claims; no independent source consultation or mathematical verification is asserted here.