Equality of Refined Tropical and Algebraic Curve Counts

Definitions and mathematical statement

Let $y$ denote the refinement parameter, and write $m_{\mathrm{BG}}(\Gamma;y)$ for the Block--Göttsche refined tropical curve-counting multiplicity referred to in the supplied record. The record also refers to a refined curve-counting polynomial $N_{g-\delta}(y)$, but supplies neither its construction nor definitions of the indices $g$ and $\delta$.

The proposed assertion is that, for every tropical curve $\Gamma$ in the intended class, \[ m_{\mathrm{BG}}(\Gamma;y)=N_{g-\delta}(y), \] with a compatible specialization \[ m_{\mathrm{BG}}(\Gamma;1)=N_{g-\delta}(1). \] These displays express the supplied target, not an established equality. The intended class of tropical curves, the hypotheses under which their multiplicities are defined, the algebraic curve-counting data associated to each $\Gamma$, and the coefficient ring of the proposed identity are unspecified. In particular, the right-hand side has no explicit dependence on $\Gamma$ in the supplied notation; its intended dependence must be recovered before this can be a fully quantified mathematical statement. The record also does not identify the ordinary invariant that the specialization at $y=1$ must recover. Thus "specializing correctly" includes an unresolved requirement beyond the displayed specialized equality.

Short English statement

The question asks whether each eligible tropical curve's Block--Göttsche refined multiplicity has the proposed algebraic interpretation through $N_{g-\delta}(y)$, including the intended ordinary specialization at $y=1$. The eligible curves, the polynomial, its indices, and the correspondence between the tropical and algebraic data require clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003017$, title: Equality of Refined Tropical and Algebraic Curve Counts; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and records proposed year 2015; no individual proposer is supplied.
• Supplied report citation: Non-Archimedean Geometry and Applications (2016), Oberwolfach Reports, pp. 3271--3331, DOI: https://doi.org/10.4171/owr/2015/57. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. The citation's 2016 date and the DOI's 2015 component are retained as supplied.
• Nicaise, Payne, and Schroeter, supplied literature reference: https://msp.org/gt/2018/22-6/gt-v22-n6-p02-s.pdf.
• Bousseau, supplied reference labelled "higher-genus refined curve counting": https://doi.org/10.1007/s00222-018-0823-z.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports that the Block--Göttsche geometric interpretation is proved for genus one, for the first low-codimension cases, and after specialization to ordinary Euler characteristic, while higher-genus geometric equalities remain only partly established. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. That supplied editorial label does not resolve the missing definitions identified in this draft or constitute independent mathematical verification.