Answered Definition of Burniat Surfaces

Definitions and mathematical statement

For a complex algebraic surface $S$, let ${\mathbb{C}}(S)$ denote its field of rational functions. A smooth projective surface is minimal if it contains no smooth rational curve of self-intersection $-1$.

The supplied record asks, "What are Burniat surfaces?" It then describes a Burniat surface as the minimal model associated with a specified nine-line configuration in $\mathbb{P}^{2}$, with the following displayed function-field expression: \[ {\mathbb{C}}\left(\sqrt{\Delta_1/\Delta_2},\sqrt{\Delta_1/\Delta_3}\right). \] This expression is retained as supplied; it does not specify a complete construction. The nine-line configuration and the meanings of $\Delta_1,\Delta_2,\Delta_3$ are absent. In particular, it is unclear whether these symbols denote individual line equations or products of equations for groups of lines.

If the intended construction is a finite cover of the complex projective plane, its function field would need to be specified as an extension of ${\mathbb{C}}(\mathbb{P}^{2})$. Conditional on $\Delta_1/\Delta_2$ and $\Delta_1/\Delta_3$ being well-defined nonzero rational functions on that plane, the corresponding candidate expression is \[ {\mathbb{C}}(\mathbb{P}^{2})\left(\sqrt{\Delta_1/\Delta_2},\sqrt{\Delta_1/\Delta_3}\right). \] This is an explicitly conditional interpretation, not a correction verified against the report. The supplied verification describes the construction as a bidouble cover, but the missing configuration and equations prevent checking its degree and geometric hypotheses. No separate unresolved mathematical assertion or quantified conjecture is supplied.

Short English statement

The record gives an answered definitional question about Burniat surfaces, described through a nine-line configuration and a bidouble-cover construction. The configuration, the defining expressions, and the base function field are insufficiently specified to recover a precise construction. No distinct open problem can be formulated from the supplied text.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001575$, source code catalogue entry, title "Answered Definition of Burniat Surfaces." No individual proposer is supplied; the recorded proposed year is 2010.
• Supplied report citation: Komplexe Analysis (2011), Oberwolfach Reports, pp. 2283--2333, DOI 10.4171/owr/2010/39. Supplied source URL and literature reference labelled "original OWR report": https://doi.org/10.4171/owr/2010/39.
• Literature assessment dated exactly 2026-08-21: the supplied curation classifies the record as open conservatively because the extracted text is a definitional question with no distinct claim to verify. This is supplied dated curation, not an independently verified literature assessment.
• Supplied statement-verification note: the heading asks what Burniat surfaces are and the following text answers with a construction from nine-line configurations as bidouble covers. The record marks the statement unrecoverable, promotes no separate clean statement, and retains the original for provenance. The cited report has not been independently consulted for this draft.