Generic Finiteness of K3 Severi Moduli Maps

Definitions and mathematical statement

Under the usual Severi-variety interpretation, let $(X,L)$ be a smooth projective polarized $K3$ surface: $\omega_X\simeq\mathcal O_X$, $H^1(X,\mathcal O_X)=0$, and $L$ is ample. The supplied record does not specify the base field, whether $L$ is primitive, or the polarization moduli space relative to which $(X,L)$ is general; these remain unresolved hypotheses.

The conventional notation is as follows. Write $L^2=2p-2$. For a positive integer $n$ and a nonnegative integer $\delta$, let $V_{n,\delta}(X)$ denote the locus in $|L^{\otimes n}|$ parametrizing integral curves with exactly $\delta$ ordinary nodes and no other singularities. Their normalizations have genus \[ g=1+n^2(p-1)-\delta. \] These conventions explain the notation but are not explicitly supplied in the record and require confirmation against the report. In particular, the intended admissible ranges of $p,n,\delta$, including any restrictions on $g$, are not specified.

For a nonempty irreducible component $V\subset V_{n,\delta}(X)$, the normalization moduli map, under the usual coarse-moduli interpretation of $M_g$, is \[ \psi_{X,n,\delta}|_V:V\longrightarrow M_g, \qquad C\longmapsto[\widetilde C], \] where $\widetilde C$ is the smooth normalization of $C$, and $M_g$ parametrizes smooth connected projective curves of genus $g$. Let $Z$ be the reduced closure of its image. Generically finite onto the image means that the induced function-field extension $k(Z)\hookrightarrow k(V)$ is finite; birational onto the image means that this inclusion is an isomorphism.

Question as supplied. For a general polarized $K3$ surface $(X,L)$ and an irreducible component $V\subset V_{n,\delta}(X)$, is $\psi_{X,n,\delta}|_V$ generically finite, or birational when $g>1$, onto its image in $M_g$? The accompanying assessment describes the intended scope as all components and all admissible $p,n,\delta$. The source wording does not clarify whether "or" distinguishes two proposed assertions or requests generic finiteness together with the stronger birationality assertion for $g>1$. No unspecified low-genus or numerical hypotheses are added here.

Short English statement

For curves with a fixed number of nodes on a general polarized $K3$ surface, ask whether the isomorphism class of the smooth normalization determines only finitely many curves in each Severi component, and whether it determines a unique general curve birationally when $g>1$. The precise parameter ranges, base field, polarization conventions, and relationship between these two assertions require clarification.

Sources

• Local UnsolvedMath registry, permanent ID $30002977$, title Generic Finiteness of K3 Severi Moduli Maps; exact source problem code catalogue entry. The supplied collection is Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2015.
• Supplied report attribution: 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 citation's 2016 date is retained alongside the supplied proposed year 2015 and the DOI containing 2015; their relationship has not been independently checked.
• Ciliberto--Flamini--Galati--Knutsen, Moduli of Nodal Curves. https://doi.org/10.1016/j.aim.2017.01.021.
• The Moduli of Singular Curves on K3 Surfaces. https://arxiv.org/abs/1401.1047.
• Curves of Maximal Moduli on K3 Surfaces. https://doi.org/10.1016/j.jms.2021.104832.
• Supplied literature assessment, checked 2026-08-21: labeled partially solved, with generic finiteness and birationality reported in many ranges, while the question for all components and all $p,n,\delta$ is reported to remain open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report. This verification claim is supplied metadata and has not been independently confirmed for this draft.