k-Differential Interpretation of Leaky Double Hurwitz Numbers

Definitions and mathematical statement

For a smooth connected compact complex curve $C$ of genus $g\in{\mathbb{Z}}_{\geq 0}$, a nonzero meromorphic $k$-differential, with $k\in{\mathbb{Z}}_{>0}$, is a nonzero meromorphic section $\eta$ of $K_C^{\otimes k}$, where $K_C$ is the canonical line bundle. If its zeros and poles are supported at distinct marked points $p_1,\ldots,p_N$, write \[ \operatorname{div}(\eta)=\sum_{i=1}^{N}m_i p_i, \qquad m_i\in{\mathbb{Z}}, \qquad \sum_{i=1}^{N}m_i=k(2g-2). \] Positive and negative orders describe zeros and poles, respectively. This supplies standard terminology on smooth complex curves; the source does not specify the intended ground field, compactification, or treatment of singular curves. Moreover, the source profile $A$ cannot simply be identified with the order vector $(m_i)$: its displayed genus-zero example uses shifted orders.

Problem. For arbitrary genus $g$, positive integer $k$, and an admissible zero--pole profile $A$ in the source's intended convention, formulate a natural enumerative problem involving $k$-differentials subject to prescribed period or residue conditions, and determine its relationship to the $k$-leaky double Hurwitz number $H_g^k(A)$. The intended goal is an enumerative interpretation for $k>0$, extending the genus-zero similarities reported in the source. No specific equality, normalization, or correspondence is supplied.

The source gives a genus-zero example with $k=1$ and \[ A=(d,-b_1,\ldots,-b_n),\qquad b_j\geq 0, \qquad d=n-1+\sum_{j=1}^{n}b_j. \] Its displayed divisor formula has an indexing defect. If the listed $b_j$ are paired in order with $p_{j+1}$, the formula reads \[ \operatorname{div}(\eta) =(d-1)p_1-\sum_{j=1}^{n}(b_j+1)p_{j+1}. \] Under this explicitly conditional reindexing, the total order is $-2$, as required for a meromorphic one-form in genus zero. The coefficients must be integers for this to be a divisor; the excerpt states nonnegativity of the $b_j$ but does not explicitly state integrality. The source labels the associated locus by the flattened notation $\texttt{DR00 (A)}$, whose intended typography and ambient moduli space are not specified.

Unresolved formulation requirements. A precise counting problem still requires the admissibility convention for $A$, a definition and normalization of $H_g^k(A)$, the period or residue constraints and their allowed values, and the moduli space, equivalence relation, and counting weights. In particular, scalar multiples of a differential must be accounted for. For $k=1$, residues and periods have their usual meanings for meromorphic one-forms, with periods taken along appropriate cycles avoiding poles. For $k>1$, the intended notions of periods and residues require additional specification. The supplied excerpt does not give the genus-zero residue conditions, the resulting count, or a general-genus conjectural formula. Consequently this is a request to formulate an enumerative interpretation, rather than a fully specified numerical identity.

Short English statement

The problem asks for a counting interpretation of $H_g^k(A)$ using meromorphic $k$-differentials with prescribed periods or residues in arbitrary genus. The supplied source reports genus-zero similarities, but leaves the precise counting conditions, profile convention, and proposed relationship unspecified. Its partially solved label is supplied dated curation, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, permanent ID $30006030$, source problem code catalogue entry, entitled k-Differential Interpretation of Leaky Double Hurwitz Numbers. The supplied record lists proposed year 2024 and gives no named proposer.
• Supplied report citation: Riemann Surfaces: Random, Flat, and Hyperbolic Geometry (2025), Oberwolfach Reports, pp. 2369--2424, DOI 10.4171/owr/2024/41. Report link, also listed among the supplied literature sources: https://doi.org/10.4171/owr/2024/41.
• Supplied literature source labelled Cavalieri--Markwig--Schmitt: https://doi.org/10.1017/fms.2025.26. No article title or specific theorem is supplied in the record.
• Supplied literature source labelled one-part leaky covers: https://arxiv.org/abs/2509.04335. No authors or specific theorem are supplied in the record.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem partially solved and reports that genus-zero residue-counting problems and structural comparisons with $k$-leaky numbers are known, while the arbitrary-genus enumerative problem and proposed relationship remain unproved. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The registry marks the statement corrected_verified, explaining that only formula delimiters, notation, grammar, and typography were repaired. This is a supplied editorial designation and does not establish mathematical verification of this draft.