Open conjectures of Rogalski–Sierra–Stafford: the minimal model programme for elliptic algebras

The 2020 paper 'Some noncommutative minimal surfaces' by Rogalski, Sierra and Stafford proves that the third Veronese of the 3-dimensional Sklyanin algebra and the second Veronese of Van den Bergh's quadrics are minimal models in a very strong sense: they admit no nontrivial connected graded noetherian overrings with the same graded quotient ring (up to Veronese regrading), and, under a hereditary hypothesis on the degree-zero localisation, every noetherian overring of an elliptic algebra is obtained by contracting finitely many line modules. The paper explicitly leaves open the completion of this noncommutative minimal model programme: (i) whether these theorems extend to all elliptic algebras without the homological hypotheses, so that every elliptic algebra has a minimal model obtained by contracting finitely many line modules; (ii) Conjecture 7.6, that finitely graded maximal orders between a minimal elliptic surface — and similarly between a Sklyanin algebra or a Van den Bergh quadric — and its graded quotient ring are automatically noetherian; (iii) Conjecture 8.13, that any two elliptic algebras with the same graded quotient ring are related, up to isomorphism, by a finite sequence of noncommutative blowups and blowdowns; and (iv) the expectation of Remark 7.9 that minimal models birational to a Sklyanin algebra are exactly the endomorphism rings of reflexive right ideals. All of these remain open in general: Hipwood (2022) settled the classification of maximal orders and their automatic noetherianity inside the Sklyanin algebra itself, and Rogalski–Sierra–Stafford (2023) connected Sklyanin algebras and Van den Bergh quadrics by explicit blowup/blowdown sequences, but the general conjectures remain unresolved.

Background

A connected graded (cg) $k$-algebra is a finitely generated $\mathbb{N}$-graded algebra $R=\bigoplus_{n\ge0}R_n$ with $R_0=k$ over the algebraically closed field $k$. The Gelfand–Kirillov dimension (GK dimension) of an algebra is a growth measure playing the role of Krull dimension for noncommutative rings; a cg noetherian domain of GK dimension $3$ is regarded as the coordinate ring of a noncommutative projective surface, via the quotient category $\mathrm{qgr}\text{-}R$ of graded modules modulo finite-dimensional ones. Inverting all nonzero homogeneous elements of such a domain yields its graded quotient ring $Q_{\mathrm{gr}}(R)$, whose degree-zero part is a division ring; two domains are birational when these division rings agree. Artin's 1997 conjecture asserts that, birationally, noncommutative surfaces form a short list: the division ring of rational functions of any cg noetherian GK-dimension-$3$ domain is either finite over its centre, an Ore extension (a generalised skew polynomial ring) of the function field $k(C)$ of an algebraic curve $C$, or the division ring associated to a Sklyanin algebra — the generic case. This conjecture remains completely open, which motivates studying each expected birational class separately; the generic case is the class of the 3-dimensional Sklyanin algebra $S$, a graded domain on three generators with three quadratic relations, depending on a point and an automorphism of an elliptic curve.

The natural framework for algebras birational to $S$ is the class of elliptic algebras: cg domains $T$ with a distinguished central homogeneous element $g$ such that $T/gT$ is a twisted homogeneous coordinate ring $B(E,M,\tau)$ — the section ring $\bigoplus_n H^0(E,M\otimes M^\tau\otimes\cdots)$ of an ample line bundle $M$ on an elliptic curve $E$ twisted by an infinite-order automorphism $\tau$. Both $S^{(3)}$ (the third Veronese subring, i.e. the subring of degrees divisible by $3$) and the second Veronese of Van den Bergh's quadrics $Q_{\mathrm{VdB}}$ — a family of three-generator cg algebras built from the same elliptic-curve data, giving noncommutative analogues of $\mathbb{P}^1\times\mathbb{P}^1$ and of the Hirzebruch surface $\mathbb{F}_2$ — are elliptic. Noncommutative analogues of classical surface operations exist in this setting: blowing up a point of $E$ produces a subring (Rogalski; Van den Bergh), and a line module — a cyclic graded module with the Hilbert series $(1-t)^{-2}$ of $k[x,y]$ — with self-intersection $(L\cdot L)=\sum_{i\ge0}(-1)^{i+1}\dim\mathrm{Ext}^i_{\mathrm{qgr}\text{-}T}(L,L)=-1$ can be contracted (blown down) to a larger elliptic algebra, in analogy with Castelnuovo's contraction of $(-1)$-curves. A maximal order is the noncommutative analogue of an integrally closed domain: a graded Goldie domain $U$ such that any graded $V$ with $U\subseteq V\subset Q_{\mathrm{gr}}(U)$ and $aVb\subseteq U$ for some nonzero homogeneous $a,b$ equals $U$.

In the commutative theory of surfaces, the minimal model programme rests on three facts: a smooth surface is minimal iff it contains no $(-1)$-curves; every birational morphism factors into finitely many point blowups; and every surface admits a minimal model. Rogalski–Sierra–Stafford (2020) established the noncommutative counterparts of the first two facts only under homological hypotheses: they proved that $S^{(3)}$ and $Q_{\mathrm{VdB}}^{(2)}$ are minimal elliptic surfaces — they have no cg noetherian overring between them and their localisation $T(g)$ — and that for an elliptic algebra $T$ whose degree-zero localisation $T^\circ=T[g^{-1}]_0$ is hereditary (finite global dimension $1$), every noetherian overring $T\subseteq R\subset T(g)$ differs by a module-finite extension from one obtained by contracting finitely many line modules, and every graded maximal order overring of a minimal elliptic surface is just a Veronese of it. Earlier, their 'Classifying orders in the Sklyanin algebra' (2015) had classified the maximal orders inside $S^{(3)}$. The paper deliberately poses the remaining steps as open problems: removing the homological hypotheses, automatic noetherianity of graded maximal orders (Conjecture 7.6), birational connectivity by blowups and blowdowns (Conjecture 8.13), and identification of all minimal models birational to $S$ with endomorphism rings $\mathrm{End}_S(P)$ of reflexive right ideals $P$ classified by a Chern-type invariant $c(P)$ (Remark 7.9).

Problem Statement

Complete the noncommutative minimal model programme for elliptic algebras as posed by Rogalski–Sierra–Stafford (2020). Let $k$ be an algebraically closed field and let $T$ be an elliptic algebra, i.e. a cg domain with a distinguished central homogeneous element $g$ such that $T/gT\cong B(E,M,\tau)$ is a twisted homogeneous coordinate ring of an elliptic curve, with $\tau$ of infinite order and of degree $\deg M\ge 2$ (the standing convention of the body of the source paper); for item 1, following the source's introduction convention under which its ultimate-goal conjecture is stated and under which the theorems it extends are proved, $T$ has degree $\deg M\ge 3$. Establish, within these degree conventions of the source and without the auxiliary homological hypotheses (such as $T[g^{-1}]_0$ hereditary or $\mathrm{qgr}\text{-}T$ smooth) used in the proved theorems:

1. (Existence of minimal models and factorisation — the paper's stated ultimate goal, extending its Theorems 1.5 and 1.9.) Every elliptic algebra $T$ of degree $\ge 3$ has a minimal model obtained by contracting finitely many line modules of self-intersection $-1$; more precisely, every noetherian cg overring $T\subseteq R\subset T(g)$ is obtained (up to a module-finite extension) from $T$ by contracting finitely many such line modules.

2. (Conjecture 7.6.) Let $T$ be a minimal elliptic surface, i.e. an elliptic algebra with no cg noetherian overring strictly between $T$ and its localisation $T(g)$, as in Definition 1.4 of the source. Then the structural theorem for graded overrings (Theorem 7.1(1) of the source) holds without assuming the overring is noetherian; equivalently, any finitely graded maximal order $T\subseteq R\subset Q_{\mathrm{gr}}(T)$ — where finitely graded means a $\mathbb{Z}$-graded $k$-algebra whose homogeneous components are all finite-dimensional over $k$ and which is nonzero in some positive degree, the source paper's weakening of the cg condition — is automatically noetherian; and similarly for overrings of the Sklyanin algebra and the Van den Bergh quadrics.

3. (Conjecture 8.13.) Any two elliptic algebras $T,T'$ with the same graded quotient ring $Q_{\mathrm{gr}}(T)=Q_{\mathrm{gr}}(T')$ are related, up to isomorphism, by a finite sequence of blowups and blowdowns (noncommutative monoidal transformations at points of the elliptic curve and contractions of line modules of self-intersection $-1$).

4. (Remark 7.9 expectation.) The minimal models birational to the Sklyanin algebra $S$ — here in the sense of Remark 7.9 of the source: a cg noetherian algebra $T^1$ containing $g$ and birational to $T$, with the property that if $R^1$ is cg noetherian and $T^1\subsetneq R^1\subset T^1(g)$, then $T^1=R^1$; this definition does not require $T^1$ to be elliptic — are exactly the endomorphism rings $\mathrm{End}_S(P)$ of reflexive right ideals $P$ of $S$ (and their analogues for $Q_{\mathrm{VdB}}$ and Veronese rings), with $\mathrm{End}_S(P)\cong\mathrm{End}_S(P')$ if and only if the discrete Chern-type invariant $c(P)$ equals $c(P')$, where $c(P)$ is the discrete invariant analogous to a second Chern class arising in the Nevins–Stafford classification of reflexive right ideals of $S$ through formal moduli spaces, as cited in Remark 7.9 of the source [7].

These four components are one coherent objective: a full minimal model programme, in the sense of the commutative theory of surfaces, for the graded algebras constituting the generic birational class of noncommutative surfaces. Each is a separately stated open claim that may be pursued and settled on its own, and together the four constitute the stated programme. Apart from item 2, which follows the source's Conjecture 7.6 in ranging over minimal elliptic surfaces (and overrings of Sklyanin algebras and Van den Bergh quadrics) rather than arbitrary elliptic algebras, proofs must cover all elliptic algebras in the sense defined above, at the degree conventions of the source ($\deg M\ge 3$ for item 1 and $\deg M\ge 2$ for items 2–4); adding restrictions on the singularities of the localisation $T[g^{-1}]_0$, the characteristic of $k$, or the automorphism $\tau$ (beyond the defining infinite-order condition), or any further degree restriction beyond the source's conventions, would not answer the stated problem.

References

1. D. Rogalski, S. J. Sierra, J. T. Stafford, Some noncommutative minimal surfaces, Advances in Mathematics 369 (2020), 107151. DOI: 10.1016/j.aim.2020.107151; arXiv:1807.09889; https://doi.org/10.1016/j.aim.2020.107151
2. M. Artin, Some problems on three-dimensional graded domains, in: Representation Theory and Algebraic Geometry (Waltham, MA, 1995), London Mathematical Society Lecture Note Series 238, Cambridge University Press, Cambridge, 1997, pp. 1–19. DOI: 10.1017/cbo9780511525995.002; Zbl 0888.16025; https://doi.org/10.1017/cbo9780511525995.002; https://zbmath.org/?q=an%3A0888.16025
3. D. Hipwood, Maximal orders in the Sklyanin algebra, Journal of Algebra 602 (2022), 555–598. DOI: 10.1016/j.jalgebra.2022.02.015; arXiv:2008.06947; https://doi.org/10.1016/j.jalgebra.2022.02.015
4. D. Rogalski, S. J. Sierra, J. T. Stafford, Ring-theoretic blowing down II: Birational transformations, Journal of Noncommutative Geometry 17 (2023), no. 3, 1045–1097. DOI: 10.4171/jncg/510; arXiv:2107.01991; https://doi.org/10.4171/jncg/510
5. D. Rogalski, S. J. Sierra, J. T. Stafford, Ring-theoretic blowing down I, Journal of Noncommutative Geometry 11 (2017), no. 4, 1465–1520. DOI: 10.4171/jncg/11-4-9; arXiv:1603.08128; https://doi.org/10.4171/jncg/11-4-9
6. D. Rogalski, S. J. Sierra, J. T. Stafford, Classifying orders in the Sklyanin algebra, Algebra & Number Theory 9 (2015), 2055–2119. DOI: 10.2140/ant.2015.9.2055; arXiv:1308.2213; https://doi.org/10.2140/ant.2015.9.2055
7. T. A. Nevins, J. T. Stafford, Sklyanin algebras and Hilbert schemes of points, Advances in Mathematics 210 (2007), no. 2, 405–478. DOI: 10.1016/j.aim.2006.06.009; https://doi.org/10.1016/j.aim.2006.06.009