Divisorial Images of Universal K3 Moduli Maps

Definitions and mathematical statement

Let $\psi_{p,n,\delta}$ denote the universal $K3$ moduli map named in the supplied source record, and let $g$ denote the genus parameter appearing there. The record does not define the domain, target, admissible parameters, or the relation between $g$ and $(p,n,\delta)$. Accordingly, write \[ \psi_{p,n,\delta}\colon X_{p,n,\delta}\longrightarrow Y_{p,n,\delta} \] only as placeholder notation for the intended map; this does not specify either moduli space.

For a map $f\colon X\to Y$, its image is $\operatorname{Im}(f)=\{f(x):x\in X\}$. A divisorial image is usually understood to mean that its closure is a codimension-one locus in the target. Under that interpretation, the condition would be \[ \operatorname{codim}_{Y_{p,n,\delta}} \overline{\operatorname{Im}(\psi_{p,n,\delta})}=1. \] The source record does not specify whether closure is intended, whether an actual effective divisor is required, or how components are to be treated. Higher-rank Brill--Noether theory concerns loci of vector bundles of rank at least two satisfying conditions on their spaces of sections; the relevant ranks, degrees, stability conditions, and section bounds are not specified here.

Question as supplied. Among the admissible maps $\psi_{p,n,\delta}$ with $g\leq 11$, can any image be a divisor in its intended target? Can some images $\operatorname{Im}(\psi_{p,n,\delta})$ be described by higher-rank Brill--Noether theory?

These are existence and description questions, not assertions that every such image is divisorial or has such a description. A fully specified mathematical formulation requires the missing moduli-map definitions and hypotheses from the cited report. The record also leaves unclear whether the restriction $g\leq 11$ applies to the second question.

Short English statement

The problem asks whether universal $K3$ moduli maps can have divisorial images for $g\leq 11$, and whether some images admit higher-rank Brill--Noether descriptions. The supplied record omits the definitions and parameter restrictions needed to make these questions fully precise. Its partially solved label and genus-ten assessment are supplied dated curation, requiring independent mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002978$, title Divisorial Images of Universal K3 Moduli Maps, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2015.
• Supplied report citation: Mini-Workshop: Singular Curves on $K3$ Surfaces and Hyperkähler Manifolds (2016), Oberwolfach Reports, pp. 2939--2967, DOI 10.4171/owr/2015/51. https://doi.org/10.4171/owr/2015/51. The record labels the statement corrected_verified and reports checking it against the official report; that verification is supplied metadata, not an independent verification for this draft.
• Farkas--Ortega, Higher-Rank Brill--Noether Theory. Supplied literature reference: https://arxiv.org/1102.0276.
• Moduli of Nodal Curves on K3 Surfaces. Supplied literature reference: https://arxiv.org/1502.07378.
• Knutsen--Torelli, Brill--Noether Loci on Severi Varieties. Supplied literature reference: https://arxiv.org/2411.14562.
• Literature assessment dated exactly 2026-08-21: the supplied status is partially solved. The supplied assessment states that the genus-ten $K3$ image has a higher-rank Brill--Noether realization and that related loci have been studied, while a general divisorial-image criterion for $g\leq 11$ and systematic descriptions of all images remain open. This is supplied dated curation derived from the OWR record, not independently verified; the time-sensitive status and cited claims require independent verification.