Recognition of degree-9 elliptic algebras: is every elliptic algebra of degree 9 with smooth qgr a 3-Veronese of a Sklyanin algebra?

Noncommutative projective geometry seeks to classify noetherian connected graded domains of Gelfand-Kirillov dimension 3, the algebras that behave like homogeneous coordinate rings of noncommutative surfaces. Within the birational class of the three-dimensional Sklyanin algebra, Rogalski, Sierra and Stafford developed a ring-theoretic analogue of blowing up and blowing down and reduced the recognition of noncommutative projective planes to one sharply stated open question: is every elliptic algebra of degree 9 whose noncommutative projective quotient category qgr-T is smooth isomorphic to the third Veronese of a three-dimensional Sklyanin algebra? Their Theorem 4.3 proves that explicit divisor conditions (4.1)-(4.2) on the underlying elliptic curve are sufficient, these conditions are necessary, and no counterexample is known; yet they write that they do not see how to prove that (4.1) and (4.2) always hold. The problem is recorded as open: it is not known whether every smooth degree-9 elliptic algebra is isomorphic to the third Veronese of a three-dimensional Sklyanin algebra.

Background

A central problem of noncommutative projective geometry is the classification of noncommutative surfaces. By a noncommutative projective surface one means the category qgr-R of graded right modules over a noetherian connected graded (cg) domain R, i.e. a finitely generated algebra graded $R=\bigoplus_{n\ge 0}R_n$ with $R_0=k$, modulo its Serre subcategory of finite-dimensional modules; this quotient, introduced by Artin and Zhang in the 1990s, mimics the category of coherent sheaves on the (generally nonexistent) projective scheme Proj $R$. The relevant dimension notion is Gelfand-Kirillov dimension (GK dimension), an algebraic growth rate; a cg domain of GK dimension 3 plays the role of the coordinate ring of a surface, and qgr-R is called smooth if it has finite homological dimension. A conjecture attributed to Artin and popularized in the survey of Stafford and Van den Bergh predicts that, birationally — i.e. after inverting nonzero homogeneous elements and passing to the degree-zero part of the resulting graded quotient ring, a division ring playing the role of the field of rational functions — every such surface belongs to a short list, with the generic case being the three-dimensional Sklyanin algebra $S(E,\sigma)$, the algebra on three generators with three quadratic relations attached to an elliptic curve $E$ and an automorphism $\sigma$ of infinite order, which is regarded as the generic noncommutative analogue of the polynomial ring in three variables, i.e. of $\mathbb{P}^2$.

The Rogalski–Sierra–Stafford (RSS) program studies the Sklyanin birational class through a family of algebras called elliptic algebras. A twisted homogeneous coordinate ring $B(E,M,\tau)$ is the graded ring $\bigoplus_{n\ge 0}H^0(E,M_n)$ with $M_n=M\otimes M^\tau\otimes\cdots\otimes M^{\tau^{n-1}}$, where $M^\tau$ denotes the pullback of the line bundle $M$ by $\tau$; an elliptic algebra is a cg domain $R$ together with a central homogeneous element $g\in R_1$ such that $R/gR\cong B(E,M,\tau)$ for an elliptic curve $E$, a line bundle $M$, and an automorphism $\tau$ of infinite order; its degree is $\deg M$. The commutative model (allowing $\tau=\mathrm{id}$) is the anticanonical ring of a possibly singular del Pezzo surface, whose degree is the self-intersection of the canonical class, so a degree-9 elliptic algebra is a noncommutative analogue of the homogeneous coordinate ring of $\mathbb{P}^2$. Two classes of examples dominate: the Sklyanin elliptic algebras, namely the third Veronese subrings $S^{(3)}=\bigoplus_{n\ge 0}S_{3n}$ of Sklyanin algebras (a Veronese subring keeps only the graded pieces whose degrees are multiples of a fixed $d$), and the quadric elliptic algebras $Q^{(2)}$, the second Veroneses of the Van den Bergh quadrics $Q(E,\sigma,\Omega)$ — noncommutative deformations of $\mathbb{P}^1\times\mathbb{P}^1$ depending on a second parameter $\Omega$. Smith and Van den Bergh showed that for fixed $(E,\sigma)$ the quadric $Q(E,\sigma,\Omega)$ has smooth qgr only for generic $\Omega$, with a discrete family of non-smooth exceptions.

The geometric toolkit is a noncommutative blowing up and blowing down. Van den Bergh and Rogalski defined the blowup $\mathrm{Bl}_p(R)=R(p)$ of an elliptic algebra at a point $p\in E$ as an explicit subring of the graded quotient ring; line modules, i.e. cyclic modules with the Hilbert series $(1-s)^{-2}$ of $k[x,y]$, play the role of lines, and Mori and Smith defined an intersection product on them by $(L\bullet L')=\sum_{n\ge 0}(-1)^{n+1}\dim_k\mathrm{Ext}^n_{\mathrm{qgr}-R}(L,L')$. In 'Ring-theoretic blowing down: I' the authors proved a noncommutative Castelnuovo contraction criterion: if qgr-$R$ is smooth and $R$ has a line module of self-intersection $-1$, then that line contracts, i.e. $R$ is the blowup of another elliptic algebra at a point — a direct analogue of the classical fact that $(-1)$-curves on a smooth surface can be blown down. 'Ring-theoretic blowing down II' uses this machinery to build explicit birational transformations (a smooth Van den Bergh quadric is a two-point blowup of a Sklyanin elliptic algebra; a noncommutative Cremona transform), and proves the recognition Theorem 4.3: if a degree-9 elliptic algebra $T$ with smooth qgr-$T$ and $T/gT=B(E,N,\tau)$ admits a $g$-divisible right ideal $H=\bigoplus_{n\ge 0}H^0(E,N^n(-a-b-c))$ and a $g$-divisible right overmodule $M$, $T\subseteq M\subseteq T_{(g)}$, $M=\bigoplus_{n\ge 0}H^0(E,N^n(d+e+f))$, each with endomorphism ring of the same Hilbert series as $T$ (here $g$-divisible means saturated with respect to multiplication by powers of the central element $g$, and $T_{(g)}$ denotes the Ore localization of $T$ at the homogeneous elements of $T\setminus gT$, both in the precise sense of Section 2 of that paper, and the Hilbert series of a graded module is the generating series of the dimensions of its graded components), where the points $a,b,c,d,e,f\in E$ satisfy the divisor identities (4.1) $N\cong\mathcal{O}_E(\tau^{-1}a+b+c)^{\otimes 3}$ and (4.2) $\mathcal{O}_E(\tau a+b+c)\cong\mathcal{O}_E(d+e+f)$, then $T\cong S^{(3)}$ for a Sklyanin algebra $S=S(E,\sigma)$ with $\sigma^3=\tau$. These conditions are also necessary, and the authors know no degree-9 elliptic algebra with smooth qgr that fails to be a Sklyanin elliptic algebra; nevertheless, in Remark 4.13 they state they do not see how to prove that (4.1) and (4.2) always hold. This is exactly the gap between 'every known smooth degree-9 elliptic algebra is a Sklyanin 3-Veronese' and the desired recognition theorem, and it is the bottom case of the program's broader goal (Conjecture 8.13 of 'Some noncommutative minimal surfaces') of connecting all elliptic algebras in one birational class by finite chains of blowups and blowdowns, with Sklyanin algebras and quadrics as the minimal models.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $0$. Let $T$ be an elliptic algebra of degree 9 in the sense of Rogalski–Sierra–Stafford: a noetherian connected graded $k$-domain with a central homogeneous element $g\in T_1$ such that $T/gT\cong B(E,N,\tau)$, the twisted homogeneous coordinate ring of an elliptic curve $E$ attached to a line bundle $N$ of degree 9 and an automorphism $\tau$ of infinite order. Assume that qgr-$T$ is smooth.

Prove or refute: every such $T$ is isomorphic to the third Veronese $S^{(3)}=\bigoplus_{n\ge 0}S_{3n}$ of a three-dimensional Sklyanin algebra $S=S(E,\sigma)$ (necessarily with $\sigma^3=\tau$).

By Theorem 4.3 of 'Ring-theoretic blowing down II' this is equivalent to asking whether $T$ always admits a $g$-divisible right ideal $H=\bigoplus_{n\ge 0}H^0(E,N^n(-a-b-c))$ and a $g$-divisible right module $M$ with $T\subseteq M\subseteq T_{(g)}$ (the Ore localization of $T$ at the homogeneous elements of $T\setminus gT$), $M=\bigoplus_{n\ge 0}H^0(E,N^n(d+e+f))$, each having endomorphism ring with the same Hilbert series as $T$ (the conditions of Notation 3.2 of that paper), for points $a,b,c,d,e,f\in E$ satisfying the divisor identities

$N\cong\mathcal{O}_E(\tau^{-1}a+b+c)^{\otimes 3}$

$\mathcal{O}_E(\tau a+b+c)\cong\mathcal{O}_E(d+e+f)$

i.e. whether conditions (4.1) and (4.2) of that paper always hold. Two further open strands recorded in the same paper — developing contraction and intersection theory directly for elliptic algebras with non-smooth qgr (instead of passing only through a Morita context to a smooth quadric, i.e. a linking pair of bimodules relating the two rings' module categories), and extending the quadric-to-plane birational transformations beyond characteristic zero — are documented in the source paper and are not part of this problem's completion boundary.

References

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