Kodaira Dimension of K3 Severi Images

Definitions and mathematical statement

Let $M_g$ denote the moduli space of smooth, connected, projective curves of genus $g$. A $K3$ surface is a smooth, projective, geometrically connected surface $S$ with trivial canonical bundle and $H^1(S,\mathcal O_S)=0$. In the usual Severi construction, an integral nodal curve $C\subset S$ determines its smooth normalization $\widetilde C$; when this normalization has genus $g$, the associated moduli map sends the parameter representing $(S,C)$ to $[\widetilde C]\in M_g$. The supplied record does not specify the base field, polarization, curve class, number of nodes, or precise Severi parameter space.

Write $\mu\colon V\dashrightarrow M_g$ for the intended moduli map restricted to an irreducible component $V$, and let $Z_V=\overline{\mu(V)}\subset M_g$, where the image is taken on the domain of definition. The question is: \[ \text{For each component }V\text{ satisfying the intended maximal-rank condition, determine }\kappa(Z_V). \] Here the maximal-rank condition is retained as a source qualification: the record does not specify whether it refers to the generic differential rank, maximal image dimension, or another rank convention, nor does it give the comparison dimensions.

If ordinary birational Kodaira dimension in characteristic zero is intended, choose a smooth projective model $X_V$ of the function field of $Z_V$ and interpret the target as $\kappa(X_V)=\kappa(K_{X_V})$. For a smooth projective variety $X$, this invariant is $-\infty$ if $H^0(X,mK_X)=0$ for every integer $m\geq1$; otherwise it is the maximum dimension of the images of the rational maps defined by the nonzero pluricanonical systems $|mK_X|$. This interpretation is conditional: the record does not specify the field or distinguish ordinary Kodaira dimension of a projective model from logarithmic Kodaira dimension of the possibly nonproper image. These missing choices prevent a uniquely specified numerical problem.

Short English statement

Determine the Kodaira dimension of the locus in $M_g$ obtained by normalizing curves represented by a maximal-rank component of a $K3$ Severi moduli map. The supplied record leaves the parameter space, genus range, base field, rank convention, and treatment of the image boundary unspecified; these require mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002979$, source problem code catalogue entry, title "Kodaira Dimension of K3 Severi Images," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 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 citation gives 2016, while the DOI and registry year metadata contain 2015; this discrepancy is retained without resolution.
• Supplied literature reference: Moduli of Nodal Curves on K3 Surfaces. https://doi.org/10.1016/j.aim.2017.01.021.
• Supplied literature reference: Degeneration of Differentials and Moduli. https://arxiv.org/1701.07224.
• Supplied literature reference labelled "Original Kodaira-Dimension Question." https://ems.press/content/serial-article-files/46597.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that some dimensions and birational behavior of $K3$ images are known, but that no general computation of the requested Kodaira dimensions has been established. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft.