Extending the order–stack dictionary to non-hereditary orders and singular stacky curves arising from central curves on noncommutative surfaces

For an order of global dimension $2$ on a smooth quasiprojective surface over an algebraically closed field of characteristic $0$ , restricting to a central curve corresponds, under the known dictionary between orders and Azumaya algebras on stacky surfaces, to a fiber product of the curve with the associated stacky surface. Baumann–Belmans–van Garderen used this to classify noncommutative conics and skew cubics, but the dictionary they establish is largely limited to hereditary orders and smooth stacky curves. This problem asks to complete the one-dimensional dictionary: determine precisely how order-theoretic properties of the restricted order (hereditary, Bass, Gorenstein, tiled, nodal) correspond to stack-theoretic features of the resulting possibly singular stacky curve (root stack structure along possibly non-reduced divisors, singular quotient charts, presence and type of generic stabiliser), to extend the hereditariness–transversality equivalence from terminal to arbitrary orders, to decide whether the Brauer group vanishes for these singular stacky curves so that the appearing Azumaya algebras split, and to classify the sheaves of algebras produced by curves contained in or non-transversely meeting the ramification divisor.

Background

An order over a normal variety $S$ is a sheaf of algebras $\mathcal{A}$ that is coherent as a sheaf of $\mathcal{O}_S$ -modules, with center $\mathcal{O}_S$ , and generically a matrix algebra over the function field; it is a 'noncommutative space' lying finite over a commutative one. An order is tame if its ramification indices along the ramification divisor are coprime to the characteristic; over the algebraically closed field of characteristic $0$ fixed below every ramification index is invertible, so every order is tame and the tameness condition is automatic here, the substantive hypotheses being global dimension $2$ , hereditariness, and terminality. An order is hereditary if every submodule of a projective module is projective (homological dimension $\leq 1$ ); Bass, Gorenstein, tiled, and nodal orders are further local homological/combinatorial conditions on such orders (Bass implies Gorenstein, while tiled and nodal are incomparable in general), defined in the standard literature on orders, and a terminal order is a highest-order 'minimal model' in the sense of the minimal model program for orders developed by Chan–Ingalls. An Azumaya algebra is the noncommutative analogue of a vector bundle: a sheaf of algebras locally étale-locally isomorphic to a matrix algebra. A Deligne–Mumford (DM) stack is a space locally modelled by quotient charts $[U/G]$ with finite group stabilisers, and a stacky curve is a one-dimensional DM stack with finite stabilisers; the root stack $\sqrt[e]{(X,D)}$ adjoins an $e$ -th root of an effective divisor $D$ , and a curve can have a generic stabiliser (stabiliser constant along an irreducible component) or non-reduced stacky structure along a divisor. The Brauer group $\mathrm{Br}(X)$ is the group of Morita-equivalence classes of Azumaya algebras; an Azumaya algebra is split when its class is trivial, and Grothendieck-type vanishing results (e.g. Colliot-Thélène–Skorobogatov, building on Tsen's theorem) state that the Brauer group of a proper curve over an algebraically closed field is trivial, so every Azumaya algebra on such a curve splits.

There is an established dictionary in the smooth case: hereditary orders on a smooth curve correspond to smooth stacky curves (root stacks in the ramification divisor), and Faber–Ingalls–Okawa–Satriano showed that tame orders of global dimension $2$ on a smooth surface correspond to Azumaya algebras on smooth stacky surfaces obtained as root stacks. Baumann–Belmans–van Garderen extended this to central curves: if $\mathcal{A}$ is a tame order of global dimension $2$ on a smooth quasiprojective surface $S$ with ramification divisor $\Delta$ and $C\subset S$ is a curve not contained in $\Delta$ , the restriction $\mathcal{A}|_C$ corresponds to the fiber product of $C$ with the stacky surface, a stacky curve with coarse moduli space $C$ . For $C$ smooth and $\mathcal{A}$ terminal, they prove (their Proposition 21) that $\mathcal{A}|_C$ is hereditary if and only if $C$ intersects $\Delta$ transversely in the smooth locus of $\Delta$ , in which case the stacky curve is exactly the root stack in the reduced intersection divisor with the ramification multiplicities. Their classification of noncommutative conics and skew cubics shows that 'bad' intersections produce non-hereditary orders, singular stacky curves, and even sheaves of algebras that fail to be orders, with tables of examples suggesting patterns (e.g. tiled orders appearing exactly for root stacks in possibly non-reduced divisors). A complete and general dictionary, the extension of Proposition 21 beyond terminal orders, and the splitting behaviour of the Azumaya algebras appearing on singular stacky fibres are left open.

Problem Statement

Let $S$ be a smooth quasiprojective surface over an algebraically closed field of characteristic $0$ , $\mathcal{A}$ an order of global dimension $2$ on $S$ with ramification divisor $\Delta$ , and $C\subset S$ a curve; the central-curve construction associates to the restriction $\mathcal{A}|_C$ a one-dimensional Deligne–Mumford stack (stacky curve) with coarse moduli space $C$ , together with an Azumaya algebra when $\mathcal{A}|_C$ is an order. Determine the extension of the order–stack dictionary beyond the smooth/hereditary case, specifically:

(Complete dictionary) Establish a precise, general correspondence between order-theoretic properties of the $\mathcal{O}_C$ -order $\mathcal{A}|_C$ — being hereditary, Bass, Gorenstein, tiled, or nodal — and stack-theoretic features of the associated stacky curve — root stack structure along a possibly non-reduced divisor, singular quotient charts, and the presence and type of generic stabiliser — including delimiting counterexamples where a correspondence fails.
(Hereditariness–transversality beyond terminal orders) Prove or disprove that, for arbitrary orders $\mathcal{A}$ (not only terminal ones) and integral curves $C$ , the order $\mathcal{A}|_C$ is hereditary if and only if $C$ intersects $\Delta$ transversely in the smooth locus of $\Delta$ , equivalently the associated stacky curve is the root stack in the reduced intersection divisor with the ramification multiplicities. This is the hereditary instance of the general correspondence in item 1, with the criterion made explicit.
(Brauer vanishing and splitting) Determine whether the Brauer group of the singular stacky curves arising as central curves vanishes — at least when there is no generic stabiliser — so that, in particular, the Azumaya algebra on the singular fibres (e.g. the fibres with $abc=0$ in the net of skew cubics over the skew polynomial algebra with cube-root-of-unity commutation parameters) is split; a nontrivial Brauer class on such a stacky curve would refute this expectation.
(Bad-intersection classification) Develop a structural classification of the sheaves of algebras on $C$ produced by curves contained in, or meeting $\Delta$ non-transversely, including those that fail to be orders; the tabulated examples for noncommutative conics and skew cubics are instances of this classification rather than its target.

The four sub-questions jointly constitute the paper's stated future work of “understanding the precise shape of the dictionary” for singular stacky curves, and each has independent standing as an in-scope sub-question. For each of the order-theoretic properties hereditary, Bass, Gorenstein, tiled, and nodal, the correspondence sought in item 1 consists of an explicit stack-theoretic criterion equivalent to that property, or a delimiting counterexample showing that no such criterion holds; a counterexample for one property concerns that property alone. Item 4 is a classification rather than a proposition: the structural classification covers every sheaf of algebras on $C$ produced by a curve contained in or non-transversely meeting $\Delta$ — including those that fail to be orders — of which the tabulated noncommutative-conic and skew-cubic families are instances rather than the target. The higher-dimensional equivalence conjectured in Remark 12 of the source is a separate open problem and is not part of this problem's completion boundary.

References

Thilo Baumann, Pieter Belmans, Okke van Garderen, “Central curves on noncommutative surfaces”, Trans. Amer. Math. Soc. 379 (2026), no. 5, 3537–3589; arXiv:2410.07620; DOI 10.1090/tran/9571; https://doi.org/10.1090/tran/9571; https://arxiv.org/abs/2410.07620
Eleonore Faber, Colin Ingalls, Shinnosuke Okawa, Matthew Satriano, “On stacky surfaces and noncommutative surfaces”, Trans. Amer. Math. Soc. 378 (2025), no. 7, 4509–4544; arXiv:2206.13359; DOI 10.1090/tran/9201; https://doi.org/10.1090/tran/9201; https://arxiv.org/abs/2206.13359
Timothy De Deyn, “The Azumification of orders”, arXiv:2606.05137 (2026); https://arxiv.org/abs/2606.05137
Thilo Baumann, “Categorical absorption for hereditary orders”, arXiv:2505.15230 (2025); https://arxiv.org/abs/2505.15230
Daniel Chan, Colin Ingalls, “The minimal model program for orders over surfaces”, Invent. Math. 161 (2005), no. 2, 427–452; DOI 10.1007/s00222-005-0438-z; https://doi.org/10.1007/s00222-005-0438-z
Jean-Louis Colliot-Thélène, Alexei N. Skorobogatov, “The Brauer–Grothendieck Group”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge) 71, Springer, 2021; DOI 10.1007/978-3-030-74248-5; https://doi.org/10.1007/978-3-030-74248-5