Brill–Noether Intersections of K3 Moduli Images

Definitions and mathematical statement

The supplied record asks about Brill--Noether intersections associated with a $K3$ surface, but does not define the maps $\psi_{X,n,\delta}$ and $\psi_{p,n,\delta}$, their domains, or their parameter ranges. The following therefore preserves the question with its unresolved notation explicitly qualified.

For a map $f\colon S\to T$, write $\operatorname{Im}(f)=\{f(s):s\in S\}$. If both source maps take values in an ambient moduli space denoted by $M_g$, and $M^r_{g,d}\subseteq M_g$ is the intended Brill--Noether locus, put \[ I_X=\operatorname{Im}(\psi_{X,n,\delta})\cap M^r_{g,d}, \qquad I_p=\operatorname{Im}(\psi_{p,n,\delta})\cap M^r_{g,d}. \] These expressions are conditional on specifying the maps and a common ambient moduli space; neither specification is contained in the supplied record.

Under a usual convention, $M_g$ parametrizes smooth projective connected curves of genus $g$, and the underlying set of a Brill--Noether locus is \[ M^r_{g,d} =\left\{[C]\in M_g:\text{there exists }L\in\operatorname{Pic}^{d}(C) \text{ with }h^0(C,L)\ge r+1\right\}, \] where $\operatorname{Pic}^{d}(C)$ denotes the set of isomorphism classes of degree-$d$ line bundles and $h^0(C,L)=\dim H^0(C,L)$. This is an explanatory convention, not a verified definition from the source report. It does not impose base-point freeness or require a complete linear series of dimension exactly $r$; whether the intended locus imposes such conditions is unresolved. The ground field and the use of a coarse moduli space, stack, or compactification are also unspecified.

Likewise, $X^{[d]}$ conventionally denotes the Hilbert scheme of zero-dimensional subschemes of $X$ of length $d$. This interpretation requires an appropriate surface $X$ and a nonnegative integer $d$; the record does not provide the precise hypotheses on $X$.

Source question. With the notation and hypotheses intended in the cited report, describe $I_X$ and $I_p$, and determine what consequences the first intersection $I_X$ has for the Mori theory of $X^{[d]}$.

The record does not specify the meanings of $p,n,\delta$, their relation to $g$, the polarization or generality assumptions on $X$, admissible parameter ranges, the intersection structure intended, or a particular Mori-theoretic conclusion. Consequently, it does not determine a fully quantified mathematical conjecture. These missing qualifications must be recovered before this draft can serve as a precise standalone formulation.

Short English statement

The question asks how the images of two unspecified moduli maps meet a Brill--Noether locus, and what the first intersection implies for the Mori theory of the Hilbert scheme $X^{[d]}$. Essential definitions and hypotheses are missing from the supplied record, so the formulation remains conditional and requires human mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, with catalog/problems.json as the additional registry specified by the project; permanent UnsolvedMath ID $30002980$, source problem code catalogue entry, title "Brill--Noether Intersections of K3 Moduli Images." This draft uses the supplied complete record; the registry files were not independently inspected.
• 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 registry gives proposed year 2015 and supplies no individual proposer.
• Supplied literature reference: Ciliberto--Knutsen, k-Gonal Loci on K3 Surfaces. https://doi.org/10.1016/j.jsc.2013.11.007.
• Supplied literature reference: Knutsen--Torelli, Prescribed-Ramification Brill--Noether Loci. https://arxiv.org/2411.14562.
• Supplied literature reference: Singular Curves on a K3 Surface and Linear Series. https://citeseerx.ist.psu.edu/document?doi=204c4253d30a59a7601c8db2a18d4e49dc04c964&repid=rep1&type=pdf. No authors are supplied for this entry.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem "partially solved" and states that Brill--Noether loci in Severi varieties and their relation to rational curves in Hilbert schemes are substantially developed, while the requested intersections and resulting Mori-theoretic consequences are not classified in general. This is supplied dated curation, not an independently verified assessment. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. This is a supplied verification claim; neither that report nor the literature references were visited for this draft.