Irreducibility of Severi Varieties on Polarized K3 Surfaces

Definitions and mathematical statement

A polarized K3 surface is a pair $(X,L)$, where $X$ is a smooth projective K3 surface and $L$ is an ample line bundle. Here a K3 surface has trivial canonical bundle and $H^1(X,\mathcal O_X)=0$. The polarization is primitive if its class in $\operatorname{Pic}(X)$ is not a positive integral multiple of another class. Its genus $p$ is specified by $L^2=2p-2$. Write $|nL|$ for the complete linear system of $L^{\otimes n}$.

Under the usual nodal-curve interpretation, let \[ V_{n,\delta}(X,L)=\{C\in |nL|: C\text{ is reduced and irreducible, with precisely }\delta\text{ ordinary nodes and no other singularities}\}. \] An ordinary node means a geometric singularity whose completed local ring is isomorphic to $\overline{k}[[x,y]]/(xy)$, after passage to an algebraic closure of the base field $k$. The conventional numerical range is \[ p\in{\mathbb{Z}},\quad p\geq2,\qquad n\in{\mathbb{Z}},\quad n\geq1,\qquad 0\leq\delta\leq p_a(nL)=1+n^2(p-1). \] In this interpretation the normalization of $C$ has genus $1+n^2(p-1)-\delta$. These numerical conventions and the exclusion of other singularities are made explicit here; the supplied record does not state them separately.

The supplied clean statement asks about the universal Severi parameter space \[ \mathcal V_{p,n,\delta}=\{((X,L),C):(X,L)\text{ is primitively polarized of genus }p,\ C\in V_{n,\delta}(X,L)\}/\text{isomorphism}, \] where the displayed expression describes its moduli objects, rather than specifying a scheme or stack construction. The question is whether $\mathcal V_{p,n,\delta}$ is irreducible for every intended admissible choice of $(p,n,\delta)$ for which it is nonempty.

Unresolved qualifications. The base field, characteristic, moduli construction, precise parameter range, and treatment of empty loci are not specified. The nonempty-locus formulation above makes an explicit convention whose agreement with the intended source question requires review. Moreover, the original statement asks whether $V_{n,\delta}(X,L)$ is irreducible for a general polarized K3 surface $(X,L)$, whereas the clean statement asks whether the universal space $\mathcal V_{p,n,\delta}$ is irreducible. These are distinct questions: irreducibility of the total parameter space does not by itself establish irreducibility of a general fiber. The original statement also does not explicitly impose primitivity. Both formulations are retained as unresolved alternatives; no equivalence is asserted.

Short English statement

The clean statement asks whether the space parametrizing a primitively polarized K3 surface together with an irreducible nodal curve in $|nL|$ is irreducible. The original wording instead concerns curves on a general fixed polarized K3 surface. This distinction, the base field, and the exact admissible parameters require clarification. The supplied status is partially solved; this is a dated curation label, not an independently verified assessment.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30002975$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied.
• 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 gives proposed year 2015; the citation's year 2016 and the DOI's 2015 component are retained without reconciliation.
• Supplied statement-verification metadata labels the statement corrected_verified and reports comparison with the official Oberwolfach report. This is a supplied verification claim; no independent source consultation is asserted here.
• Ciliberto--Dedieu, Irreducibility of Severi Varieties. https://arxiv.org/abs/1809.03914.
• Bruno--Lelli-Chiesa, Irreducibility of Severi Varieties. https://arxiv.org/abs/2112.09398.
• Ciliberto--Flamini--Galati--Knutsen, Moduli of Nodal Curves. https://arxiv.org/abs/1502.07378.
• Supplied literature assessment, checked on 2026-08-21: status partially solved. The supplied assessment reports irreducibility results in broad ranges, including $p\geq4\delta-3$ for primitive classes and small-node ranges for general K3 surfaces, while describing the full $\mathcal V_{p,n,\delta}$ question as open. This is supplied dated curation, not independently verified. The record expressly requests independent verification of time-sensitive status and cited claims; it does not provide the complete hypotheses or identify the precise parameter spaces covered by each cited result.