Quotient realisation of log terminal surface pairs with standard coefficients and the type-A mesh parameter for log terminal orders (Chan–Ingalls, Question 2.5)

Two-dimensional orders of finite representation type — equivalently, by the minimal model program for orders, log terminal orders — were classified by Artin via ramification data and by Reiten–Van den Bergh via Auslander–Reiten quivers; Chan–Ingalls (2021) connected both classifications through an explicit classification of pairs $(G,\eta)$ with $G\subset\mathrm{GL}_2(k)$ finite and $\eta\in H^2(G,k^*)$, computing McKay quivers and matching ramification data. Two precisely stated gaps in that dictionary remain open. First (Question 2.5 of the paper): is every log terminal pair $(\operatorname{Spec}R,\Delta)$ with standard coefficients, over an algebraically closed field of characteristic zero, of the form $(\operatorname{Spec}k[[x,y]]^G,\text{ramification divisor of }\operatorname{Spec}k[[x,y]]\to\operatorname{Spec}R)$ for a finite subgroup $G\subseteq\mathrm{GL}_2(k)$? Second: in the type-A cases of the McKay/Auslander–Reiten quiver classification where the mesh-type relations depend on a root-of-unity parameter $\zeta$, determine $\zeta$ explicitly as a function of $(G,\eta)$.

Background

An order over a normal local domain $R$ is a ring $A$, module-finite and torsion-free over its centre $R$, such that $A\otimes_R K$ is a central simple algebra over $K=\operatorname{Frac}(R)$; orders are the basic noncommutative analogues of finite covers and of commutative surface singularities. An order is Azumaya at a prime if it is locally a matrix algebra there (noncommutatively trivial), and its ramification locus is the codimension-one set where Azumaya-ness fails; each ramification curve $C$ carries a ramification index $e_C$, defined by $(\operatorname{rad}A_P)^{e_C}=PA_P$. Classification is up to reflexive Morita equivalence (Morita equivalence off a codimension-two set), and an order has finite representation type if it has only finitely many indecomposable maximal Cohen–Macaulay modules (lattices) up to isomorphism.

On the commutative side, a pair $(X,\Delta)$ (normal $X$, effective $\mathbb{Q}$-divisor $\Delta$) is log terminal (klt) if every canonical divisor $K_X+\Delta$ is $\mathbb{Q}$-Cartier and all discrepancies exceed $-1$ (on a log resolution $f:Y\to X$ one writes $K_Y=f^*(K_X+\Delta)+\sum_E a_E E_E$, and the rational coefficients $a_E$ — the discrepancies — must all be $>-1$). Coefficients of the form $1-1/e$, $e\in\mathbb{Z}_{\ge1}$, are called standard coefficients. To every normal order $A$ one associates its log surface $\operatorname{Log}(A)=(\operatorname{Spec}Z(A),\Delta_A)$ where $\Delta_A=\sum_C(1-1/e_C)C$ runs over the ramification curves. The minimal model program for orders (Chan–Ingalls 2005) and its extension (Chan–Hacking–Ingalls 2009) show: $A$ has finite representation type $\iff$ $A$ is log terminal $\iff$ $\operatorname{Log}(A)$ is a log terminal pair. Classically, over an algebraically closed field of characteristic zero, a normal surface singularity is klt exactly when it is a quotient singularity $k[[x,y]]^G$; this settles the boundary-free case but says nothing about matching a prescribed boundary $\Delta$.

The classification landscape: Artin (1987) classified two-dimensional maximal orders of finite representation type by ramification data; Reiten–Van den Bergh (1989) classified normal tame orders of finite representation type by their Auslander–Reiten quivers — the AR-quiver has indecomposable lattices as vertices and irreducible maps as arrows, is for finite representation type a translation quiver $\mathbb{Z}\Delta/\langle\rho\rangle$ with $\Delta$ extended Dynkin (one of the affine diagrams $\tilde A_n$, $\tilde D_n$, $\tilde E_6$, $\tilde E_7$, $\tilde E_8$), and comes with mesh-type relations on its path algebra encoding the Auslander–Reiten translate. They also showed every tame order of finite representation type is reflexively Morita equivalent to a crossed product $k[[x,y]]*_{\eta}G$ for a finite subgroup $G\subset\mathrm{GL}_2(k)$ and a cohomology class $\eta\in H^2(G,k^*)$ (a crossed product is a 2-cocycle twist of the skew group ring — the ring generated by $k[[x,y]]$ and $G$ with $g\,m=g(m)\,g$ — in which the multiplication of group elements is deformed by $\eta$).

Chan–Ingalls (2021) made this dictionary explicit: they classified the pairs $(G,\eta)$ using the classical classification of finite subgroups of $\mathrm{GL}_2$, computed $H^2(G,k^*)$ in all cases, computed the McKay quivers of $(G,\eta)$ (vertices the irreducible representations of $G$, arrows given by tensoring with the natural two-dimensional representation, twisted by $\eta$) — these coincide with the AR-quivers — classified log terminal pairs over a smooth centre numerically (ramification along a single smooth curve with arbitrary index, or curve configurations of types $A$, $D$, $E$ whose indices form a Platonic triple, i.e. $1/a+1/b+1/c>1$, including the degenerate case where one of the integers is $1$), and tabulated the ramification divisors of all Shephard–Todd reflection groups and imprimitive groups $G(mn,n,2)$. Here a pseudo-reflection is an element of $\mathrm{GL}_2$ fixing a hyperplane pointwise; a finite unitary reflection group is a finite group generated by pseudo-reflections (classified by Shephard–Todd), and by the Chevalley–Shephard–Todd theorem $k[[x,y]]^G$ is smooth exactly when $G$ is so generated; the ramification curves of $\operatorname{Spec}k[[x,y]]\to\operatorname{Spec}k[[x,y]]^G$ are precisely the images of the pseudo-reflection fixed curves. With these tables the paper matched Artin's ramification classification to Reiten–Van den Bergh's AR-quiver classification.

Two gaps in the dictionary were left explicit. The construction above shows every quotient pair $(\operatorname{Spec}k[[x,y]]^G,\text{ramification})$ is a log terminal pair with standard coefficients; the converse surjectivity — whether an abstractly given log terminal pair, possibly over a singular complete local ring, is realised by some finite linear group with the boundary exactly the ramification divisor — is Question 2.5 of the paper and is not resolved there. Separately, Reiten–Van den Bergh's classification needs the AR-quiver together with its mesh relations; Chan–Ingalls compute only the underlying quivers, and when the associated translation quiver has type $A$ the relations are either forced or depend on an additional root-of-unity parameter $\zeta$ whose dependence on $(G,\eta)$ they left as an explicitly stated open exercise.

Problem Statement

Throughout, $k$ is an algebraically closed field of characteristic zero.

Part 1 (Question 2.5 of Chan–Ingalls, stated faithfully). Let $R$ be a complete local $k$-algebra of dimension two and let $\Delta$ be a divisor on $\operatorname{Spec}R$ with standard coefficients, i.e. each coefficient has the form $1-1/e$ with $e\in\mathbb{Z}$, $e\ge1$. Suppose $(\operatorname{Spec}R,\Delta)$ is log terminal. Does there always exist a finite subgroup $G\subseteq\mathrm{GL}_2(k)$ such that $R\simeq k[[x,y]]^G$ and $\Delta$ is the ramification divisor of the quotient map $\operatorname{Spec}k[[x,y]]\to\operatorname{Spec}R$?

An affirmative answer must hold for all such pairs, including singular centres $R$ (which are then quotient singularities) and all standard-coefficient boundaries; a negative answer must exhibit an explicit log terminal pair $(\operatorname{Spec}R,\Delta)$ and prove that no finite $G\subseteq\mathrm{GL}_2(k)$ with $R\simeq k[[x,y]]^G$ has $\Delta$ as its ramification divisor.

Part 2 (the type-A mesh parameter). In the classification of the crossed-product orders $k[[x,y]]*_{\eta}G$, Reiten–Van den Bergh determine the orders by the AR-quiver together with mesh-type relations on its path algebra. When the associated translation quiver is not of type $A$, these relations are unique up to isomorphism; in type $A$ they are either uniquely determined or depend on a root-of-unity parameter $\zeta\in k$. Determine $\zeta$ explicitly as a function of the pair $(G,\eta)$.

The common objective of the two parts is to complete the two-way dictionary between (i) log terminal surface pairs with standard coefficients (ramification data), (ii) crossed-product data $(G,\eta)$ up to the equivalences of the classification, and (iii) Auslander–Reiten quivers with their mesh relations, for two-dimensional orders of finite representation type. Part 1 supplies the realisation direction of this dictionary and Part 2 the type-A mesh relations.

References

1. Daniel Chan, Colin Ingalls, Low dimensional orders of finite representation type, Mathematische Zeitschrift 297 (2021), 1161–1190. DOI 10.1007/s00209-020-02552-2, https://doi.org/10.1007/s00209-020-02552-2 (preprint arXiv:1805.09356, https://arxiv.org/abs/1805.09356)
2. Idun Reiten, Michel Van den Bergh, Two-dimensional tame and maximal orders of finite representation type, Memoirs of the American Mathematical Society 80 (1989), no. 408. DOI 10.1090/memo/0408, https://doi.org/10.1090/memo/0408
3. Michael Artin, Two-dimensional orders of finite representation type, Manuscripta Mathematica 58 (1987), 445–471. DOI 10.1007/BF01277604, https://doi.org/10.1007/BF01277604
4. Daniel Chan, Paul Hacking, Colin Ingalls, Canonical singularities of orders over surfaces, Proceedings of the London Mathematical Society (3) 98 (2009), 83–115. DOI 10.1112/plms/pdn025, https://doi.org/10.1112/plms/pdn025
5. Daniel Chan, Colin Ingalls, The minimal model program for orders over surfaces, Inventiones mathematicae 161 (2005), 427–452. DOI 10.1007/s00222-005-0438-z, https://doi.org/10.1007/s00222-005-0438-z
6. Geoffrey C. Shephard, John Arthur Todd, Finite unitary reflection groups, Canadian Journal of Mathematics 6 (1954), 274–304. DOI 10.4153/cjm-1954-028-3, https://doi.org/10.4153/cjm-1954-028-3
7. Manuel L. Reyes, The finite dual coalgebra as a quantization of the maximal spectrum, Journal of Algebra 644 (2024), 287–328. DOI 10.1016/j.jalgebra.2023.12.035, https://doi.org/10.1016/j.jalgebra.2023.12.035 (preprint arXiv:2111.07081, https://arxiv.org/abs/2111.07081)
8. Daniel Chan, Colin Ingalls, Some terminal orders on 3-folds, preprint (2026), arXiv:2607.02950, https://arxiv.org/abs/2607.02950