Dual Graphs of Negative Curves on Enriques Surfaces

Definitions and mathematical statement

The Enriques lattice is conventionally written \[ L_{\mathrm{Enr}}=U\oplus E_8(-1), \] where $U$ is the rank-two lattice with Gram matrix $\begin{pmatrix}0&1\\1&0\end{pmatrix}$, and $E_8(-1)$ is the negative-definite $E_8$ root lattice. A root is an element $r\in L_{\mathrm{Enr}}$ satisfying $(r,r)=-2$.

In the usual geometric interpretation, a $(-2)$-curve on a smooth projective surface is a smooth rational curve $C$ with self-intersection $C^2=-2$. For a finite collection of distinct such curves $C_1,\ldots,C_n$, its intersection data are \[ C_i^2=-2,\qquad m_{ij}=C_i\cdot C_j\quad(i\ne j). \] A dual graph may encode these data by vertices corresponding to the curves and intersection labels $m_{ij}$. Correspondingly, roots $r_1,\ldots,r_n$ in $L_{\mathrm{Enr}}$ encode these data when $(r_i,r_j)=m_{ij}$. This explains the customary intersection interpretation; the supplied record does not specify the intended graph conventions or the precise lattice data retained by an abstract graph.

Classification and realization problem. Classify the abstract good dual graphs in the Enriques lattice, and find Enriques surfaces realizing the corresponding $(-2)$-curve configurations.

Unresolved qualifications. The supplied record does not define the predicate "good." Consequently, it does not determine the class of graphs to be classified. It also leaves unspecified the ground field and characteristic, the intended convention for Enriques surfaces, whether only finite configurations are considered, and whether classification is up to graph isomorphism or up to lattice isometry of embedded configurations. For a fixed geometric setting and a finite graph carrying intersection labels $m_{ij}$, realization in the intersection sense means the existence of an Enriques surface $X$ and distinct $(-2)$-curves $C_1,\ldots,C_n\subset X$ satisfying $C_i\cdot C_j=m_{ij}$ for every $i\ne j$. The record does not state whether a prescribed lattice embedding must also be realized, or whether every abstract good graph is expected to be realizable. These missing qualifications prevent a fully specified quantified classification statement.

Short English statement

The problem asks for a classification of abstract good dual graphs in the Enriques lattice and for Enriques surfaces containing the corresponding configurations of $(-2)$-curves. The meaning of "good," the geometric setting, and the equivalence and realization requirements must be recovered from the source before the target can be made fully precise.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002974$, source code catalogue entry, title "Dual Graphs of Negative Curves on Enriques Surfaces," 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 registry separately supplies proposed year 2015; the report-year discrepancy is unresolved.
• Supplied literature reference: Katsura--Kondo--Martin, Enriques Surfaces in Characteristic Two. https://doi.org/10.1093/imrn/rnx074.
• Supplied literature reference: Martin, Classification of Enriques Surfaces. https://doi.org/10.46298/ag.2020.7.390.
• Supplied literature reference: On Extra-Special Enriques Surfaces. https://doi.org/10.1007/s00208-022-02459-9.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem partially solved and reports that many realizable $(-2)$-curve configurations and complete classifications in important finite-automorphism cases are known, while classification of all abstract good dual graphs in the Enriques lattice remains incomplete. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its 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, preserving the report's hypotheses, notation, and open target. This is a supplied source-verification claim, not verification performed for this draft.