Brauer Groups of Tame Stacky Curves

Definitions and mathematical statement

Let $k$ be an arbitrary algebraically closed field and let $\mathcal X$ be a tame stacky curve over $k$, with the precise class of stacky curves subject to the qualifications below.

For an algebraic stack with finite inertia, tameness means that the geometric stabilizer group schemes are linearly reductive: taking invariants in their representations is exact. The supplied record does not specify the additional hypotheses included in its use of "stacky curve," such as smoothness, separatedness, properness, or triviality of the generic stabilizer. These hypotheses must therefore remain unresolved rather than be imposed here. No restriction on $\operatorname{char}(k)$ is supplied.

The Azumaya Brauer group $\operatorname{Br}_{\mathrm{Az}}(\mathcal X)$ is the group of Morita-equivalence classes of Azumaya $\mathcal O_{\mathcal X}$-algebras, with operation induced by tensor product. An Azumaya algebra is an algebra that is locally a matrix algebra in the $\mathrm{\acute etale}$ topology. The cohomological Brauer group is \[ \operatorname{Br}'(\mathcal X) := H^2_{\mathrm{\acute et}}(\mathcal X,\mathbb G_m)_{\mathrm{tors}}, \] where $\mathbb G_m$ is the sheaf of invertible functions and the subscript denotes the torsion subgroup. The supplied record does not specify which meaning of Brauer group is intended; no equality between these groups is assumed.

Requested computation. For every algebraically closed field $k$ and every tame stacky curve $\mathcal X/k$ in the intended class, determine the Brauer group of $\mathcal X$, up to group isomorphism, in terms of specified geometric and stabilizer data. This expresses the supplied request to compute the group for an arbitrary such curve. A fully unambiguous formulation requires fixing both the class of curves and the Brauer-group convention; the record supplies neither a particular formula to establish nor a specified format for the computation.

Short English statement

Determine the Brauer group of every tame stacky curve over an arbitrary algebraically closed field. The intended hypotheses on the curve and the choice between the Azumaya and cohomological Brauer groups remain unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30005975$, source code catalogue entry, title "Brauer Groups of Tame Stacky Curves." The supplied record gives no individual proposer and records the proposed year as $2024$.
• Supplied primary-source citation: Explicit Methods in Number Theory (2025), Oberwolfach Reports, pp. 2309--2368, DOI 10.4171/owr/2024/40. https://doi.org/10.4171/owr/2024/40. The supplied background attributes the question to workshop open-problem discussions collected in Oberwolfach Reports.
• Achenjang, On Brauer groups of tame stacks. https://arxiv.org/abs/2410.06217.
• Bishop, Brauer groups of tame stacky curves. https://arxiv.org/abs/2507.08780.
• Supplied literature assessment, checked 2026-08-22: the registry labels the problem "partially solved" and states that general tools compute many tame stacky-curve Brauer groups, including broad algebraically closed-field and modular-curve cases, while the unrestricted question is not a single completed classification. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and says that only formula delimiters, notation, grammar, and typography were repaired, with mathematical content and scope preserved. These are supplied editorial claims, not independent mathematical verification of this draft.