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Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

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Pith's one-line read Feigin–Odesskii elliptic algebras map onto twisted homogeneous coordinate rings of their characteristic varieties, and the maps are surjective exactly when the characteristic variety is a power or symmetric power of the base elliptic curve.

desk verdict Strong, useful paper on Feigin–Odesskii algebras with a real proof gap in the key multiplication theorem; the results look right but Theorem 4.9 needs a corrected proof. read the letter →

arxiv 1908.06525 v3 pith:B3OGYVGK submitted 2019-08-18 math.AG math.QAmath.RA

Alex Chirvasitu , Ryo Kanda , S. Paul Smith This is my paper · ORCID
classification math.AGmath.QAmath.RA MSC 14A2216S3816W5014H5214F05
keywords ellipticalgebrasFeigin–Odesskiitwistedhomogeneouscoordinateringscharacteristicvarietycurvesymmetricpowerprojectivenormalitysemistablevectorbundles
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Feigin and Odesskii’s elliptic algebras $Q_{n,k}(E,\tau)$ are noncommutative deformations of the polynomial ring in $n$ variables, built from $\theta$ functions on an elliptic curve $E$. This paper proves that each such algebra has a canonical graded homomorphism to a twisted homogeneous coordinate ring $B(X_{n/k},\sigma',L'_{n/k})$ attached to its characteristic variety $X_{n/k}$, and that when $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$, the homomorphism is surjective. It further proves that the ideal of relations of the target ring is generated by elements of degrees 2 and 3, and that $X_{n/k}$ sits inside the noncommutative projective scheme $\mathrm{Proj}_{nc}(Q_{n,k}(E,\tau))$ as a closed subvariety. In the classical case $\tau=0$, these algebraic statements say that the complete linear system $|L_{n/k}|$ embeds $E^g$ as a projectively normal subvariety of projective space that is a scheme-theoretic intersection of quadrics and cubics. The interest is that a highly noncommutative construction is shown to have precisely the same graded ring-theoretic properties as the classical coordinate rings of abelian varieties.

What carries the argument

The load-bearing object is the canonical homomorphism $\Psi_{n/k}:Q_{n,k}(E,\tau)\to B(X_{n/k},\sigma',L'_{n/k})$, built in two steps: identify the degree-one component of $Q_{n,k}$ with $H^0(E^g,L_{n/k})$, and use a $\theta$-function identity in $g$ variables to show that the quadratic relations of $Q_{n,k}$ vanish on the graph of the automorphism $\sigma'$ of $X_{n/k}$. The proofs of surjectivity and of the relation-degree bound reduce to a question about multiplication of global sections: given locally free sheaves $U,V$ on an elliptic curve $E$, when is $H^0(E,U)\otimes H^0(E,V)\to H^0(E,U\otimes V)$ surjective? The paper proves a slope criterion (Theorem 4.9): if $U$ and $V$ are semistable, generated by global sections, and $1/\mu(U)+1/\mu(V)<1$, the multiplication map is onto. For $S^gE$, the symmetric power is viewed as a projective space bundle $P(\mathcal{E})$ over $E$, and the sheaf $L'_{n/k}$ has N\'eron–Severi class $D+(m-1)F$, putting it in the range where the slope criterion applies. For $E^g$, the argument pushes sheaves down along the projection $E^g\to E$ and runs an induction on $g$, using Grauert's theorem to keep track of the fibers.

What would settle it

Take a concrete case with all continued-fraction entries at least 3, for example $(n,k)=(8,3)$ with $8/3=[3,3]$, and compute the Hilbert function of the graded ring $\bigoplus_{m\ge0}H^0(E^2,L_{8/3}^{\otimes m})$ modulo the relations coming from degrees 2 and 3. If the degree-4 part is larger than the degree-4 part of the section ring, then the relations are not generated in degrees 2 and 3 and Theorem 9.7 fails. Alternatively, for a fixed $\tau$, directly check whether each quadratic relation in (1-3) vanishes on the graph of $\sigma'$; a single counterexample would destroy the homomorphism.

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Extended reading notes

Core claim

The paper’s central claim is Theorem 1.1: for relatively prime integers $n>k\ge 1$, the canonical homomorphism $\Psi_{n/k}: Q_{n,k}(E,\tau)\to B(X_{n/k},\sigma',L'_{n/k})$ exists and is nontrivial, and whenever the negative continued fraction for $n/k$ has all entries at least 3, or is of the form $[m,2,\ldots,2]$ or $[2,\ldots,2,m]$ with $m\ge3$, the homomorphism is surjective and the target algebra has its relations generated in degrees at most 3. In the first case $X_{n/k}\cong E^g$; in the second $X_{n/k}\cong S^gE$. Because the sheaf $L'_{n/k}$ is $\sigma'$-ample, the quotient-category equivalence for twisted homogeneous coordinate rings turns the surjectivity into the statement that $X_{n/k}$ is a closed subvariety of $\mathrm{Proj}_{nc}(Q_{n,k}(E,\tau))$. For $\tau=0$ and $X_{n/k}=E^g$, the relation-degree bound says that the image of $E^g$ in $\mathbb{P}^{n-1}$ is projectively normal and cut out by quadrics and cubics.

Load-bearing premise

The argument depends on a vanishing statement, quoted from the companion paper [CKS19b, Cor. 5.9]: the quadratic $\theta$ relations defining $Q_{n,k}(E,\tau)$ vanish on the graph of $\sigma'$; if that statement failed, there would be no homomorphism $\Psi_{n/k}$ at all, and the surjectivity and relation-degree conclusions would be moot.

Editorial extensions

If this is right

  • When $X_{n/k}\cong E^g$ or $S^gE$, the algebra $B(X_{n/k},\sigma',L'_{n/k})$ is generated by its degree-one component, equivalently $\Psi_{n/k}$ is surjective.
  • The ideal of relations of $B(X_{n/k},\sigma',L'_{n/k})$ is generated by elements of degrees 2 and 3.
  • The noncommutative scheme $\mathrm{Proj}_{nc}(Q_{n,k}(E,\tau))$ contains $X_{n/k}$ as a closed subvariety, in the sense of a fully faithful quotient-category functor whose image is closed under subquotients.
  • When $\tau=0$ and $X_{n/k}=E^g$, the morphism $\Phi_{|L_{n/k}|}:E^g\to\mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.
  • In the symmetric-power case, the same conclusions hold for every translation automorphism $\sigma$ of $S^gE$ and every ample invertible sheaf with N\'eron–Severi class $aD+bF$, $a\ge1$, $b\ge2$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same slope-based technique should apply to the remaining characteristic varieties $X_{n/k}$, which the quoted structure results describe as bundles over a power of $E$ with projective-space fibers; the paper establishes the two extreme cases where the fiber dimension is zero or maximal.
  • The $\tau=0$ consequence suggests a testable pattern: for each continued-fraction datum $n/k$, the image of $E^g$ under $\Phi_{|L_{n/k}|}$ should be projectively normal with relations generated in degrees depending on the largest entry $n_i$, perhaps exactly 2 and 3 whenever all $n_i\ge3$.
  • If the vanishing statement of Proposition 3.5 is the only input that creates the homomorphism, then the kernel of $\Psi_{n/k}$ is essentially the syzygy module of the characteristic variety; one could try to recover the minimal relations of $Q_{n,k}$ itself from the minimal free resolution of $X_{n/k}$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies the Feigin–Odesskii elliptic algebras Q_{n,k}(E,τ) (n>k≥1 coprime, E a complex elliptic curve, τ∈E). The central object is a canonical graded homomorphism Ψ_{n/k}: Q_{n,k}(E,τ) → B(X_{n/k},σ′,L′_{n/k}) to the twisted homogeneous coordinate ring of the characteristic variety X_{n/k}, together with the identification B(X_{n/k},σ′,L′_{n/k}) ≅ B(E^g,σ,L_{n/k})^{Σ_{n/k}} imported from [CKS19b]. The main theorem (Theorem 1.1) asserts: when the negative continued fraction of n/k has all entries ≥3 (so X_{n/k} ≅ E^g), or exactly one of the two end entries is ≥3 and the rest are 2 (so X_{n/k} ≅ S^gE), then Ψ_{n/k} is surjective, the relations of B are generated in degrees ≤3, and X_{n/k} occurs as a closed subvariety of Proj_nc(Q_{n,k}(E,τ)). For τ=0 and X_{n/k}=E^g the authors deduce that |L_{n/k}| embeds E^g as a projectively normal, scheme-theoretic intersection of quadrics and cubics. The technical core consists of a new surjectivity criterion for multiplication maps of sections of semistable bundles on an elliptic curve (Theorem 4.9), applied through the projective-bundle presentation of S^gE (Sections 5–6) and through new semistability results for direct images of tensor products of translates of L_{n/k} on E^g (Sections 8–9).

Significance. Assuming correctness, the paper delivers a substantial new structural result for the family Q_{n,k}(E,τ) with k>1: degree-one generation and degree-≤3 defining relations for the associated twisted homogeneous coordinate rings, in cases where the classical commutative theorems (Koizumi, Mumford, Pareschi–Popa), which concern triple powers of ample line bundles, do not apply. The τ=0 geometric corollary (projective normality and quadric–cubic generation of the ideal of E^g under |L_{n/k}|) is concrete and checkable, and the authors are careful to attribute known results (Butler, Catanese–Ciliberto, Tu). The paper is explicit and largely well structured, with the main technical results stated as Theorems 5.7, 6.9, 8.1, and 9.7. Credit should also go to the honest limitation statements: the paper identifies exactly which inputs are imported from the companion papers [CKS19b] and [CKS20]. The main caveats are that one step of the proof of Theorem 4.9 is incorrect as written, and that part of Theorem 1.1 is proved only for one of the two continued-fraction shapes; both are local and repairable.

major comments (3)
  1. [§4.2, proof of Theorem 4.9] The Serre-duality step in the proof of Theorem 4.9 is incorrect as written. The text states: "Since H^1(ε) is an isomorphism, H^0(ε^∨) is also an isomorphism by Serre duality." But H^1(E,U)=0 by Lemma 4.8(1) while H^1(E,H^0(E,U)⊗O_E) ≅ H^0(E,U) ⊗ H^1(E,O_E) ≅ H^0(E,U) ≠ 0, so H^1(ε) is the zero map, not an isomorphism. Hence the asserted identification of M := ker(H^0(E,K^∨)⊗O_E → K^∨) with U^∨ is not established by the diagram as written. The conclusion is nevertheless correct and the proof is repairable: in the dual exact sequence 0 → U^∨ → H^0(E,U)*⊗O_E → K^∨ → 0 the connecting map into Ext^1(U,O_E) ≅ H^1(U^∨) vanishes because H^1(ε^∨): H^1(U^∨) → H^1(O^N) is the Serre dual of the isomorphism H^0(ε): H^0(O^N) → H^0(U); consequently H^0(q): H^0(O^N) → H^0(K^∨) is an isomorphism for the quotient q, and the diagram chase gives M ≅ U^∨. Since Theorem 4.9 feeds Corollaries 4.10–4.11, Proposition 5.6, Lemmas 6.8 and 8.6, and Theorems 5.7, 6.9, 8.1, and 9.7 — the results behind Theorem 1.1(4)–(5) — the proof must be corrected before the main claims can be regarded as proved.
  2. [Theorem 1.1(3)–(5); §7.3 and §7.4] Theorem 1.1(3)–(5) is stated for both continued-fraction shapes that yield X_{n/k} ≅ S^gE, namely n/k = [m,2^{g−1}] and n/k = [2^{g−1},m] with m ≥ 3. However, the body proves the generation and relation-degree results only for the first shape: Lemma 7.4, Theorem 7.5, and Corollary 7.6 each explicitly assume n/k = [m,2^{g−1}], and the proof of Theorem 1.1 cites Theorem 7.5 together with Proposition 8.1 and Theorem 9.7, which concern the E^g case. Proposition 7.3 does not cover the second shape: its hypothesis (n,k) = ((m−1)g+1, g) is precisely the [m,2^{g−1}] case, not the [2^{g−1},m] case (n,k) = ((m−1)g+1, (m−1)(g−1)+1). Thus Theorem 1.1(4)–(5) for [2^{g−1},m] is not proved as written. Similarly, §1.3.1's "similar argument using Theorem 5.7" for the second shape is incomplete without a computation of [L′_{n/k}] analogous to Lemma 7.4. The authors should add the missing case, e.g. via the reversal symmetry (z_1,…,z_g) ↦ (z_g,…,z_1) (equivalently k ↦ k^{−1} mod n, which relates the two shapes), or explicitly restrict the statements.
  3. [§3.1.7 and Corollary 3.6 (Prop. 3.5)] The central input of the paper is Proposition 3.5, quoted verbatim from [CKS19b, Cor. 5.9] (the quadratic relations of Q_{n,k}(E,τ) vanish on the graph of σ′), which is the only reason the homomorphism Ψ_{n/k} of Corollary 3.6 exists; without it all subsequent surjectivity and relation-degree results are moot. Likewise Theorem 1.1(3) (X_{n/k} ≅ E^g or S^gE) is imported from [CKS19b, §§4.6.1 and Cor. 4.24] via §3.1.7, and Theorem 3.2(5) (the Σ_{n/k}-invariant isomorphism B(X_{n/k},σ′,L′_{n/k}) ≅ B(E^g,σ,L_{n/k})^{Σ_{n/k}}) is quoted from the companion paper. These are load-bearing and the present manuscript contains no proof or even a precise restatement of them. Because [CKS19b] and [CKS20] are cited as arXiv preprints, the paper cannot currently be verified independently; the authors should state the quoted results fully and clarify their publication status.
minor comments (4)
  1. [§7.4, Lemma 7.7] Lemma 7.7 assumes τ ∈ (E − E[2]) ∪ {0}, but it invokes [CKS20, Thm. 5.10] for dim Q_2, whereas §1.2 states the Hilbert-series results of [CKS20] only for non-torsion τ; since E − E[2] includes points of finite order other than 2, the scope of the cited theorem should be clarified.
  2. [§4.2] In the proof of Theorem 4.9 the symbol ε^∨ is used for two different maps: the transpose U^∨ → H^0(E,U)*⊗O_E and the quotient H^0(E,U)*⊗O_E → K^∨. This notational clash contributes to the confusion in the diagram chase; please distinguish the two maps.
  3. [§3.2] Two lines before Corollary 3.6, "Qn/k(E,τ)" should read "Q_{n,k}(E,τ)".
  4. [§9.2] The proof of Theorem 9.7 is highly condensed: the construction of the sheaf C, the claim that K(L,M)⊗π∗K and C are semistable of positive slope, and the vanishing H^1(E,K(L,G))=0 are dispatched in a few sentences. Given that this theorem completes the main claim on relations, a few more sentences explaining the semistability arithmetic (where each slope bound comes from) would considerably help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main theorems are proved from independent semistable-bundle multiplication results, and self-citations function as external dependencies rather than constructional reductions.

full rationale

The paper's central claims, Theorem 1.1(4)-(6), are established by proving that twisted homogeneous coordinate rings B(X_{n/k}, sigma', L'_{n/k}) are generated in degree one and have relations generated in degrees 2 and 3. These proofs are carried out in Sections 4-9: Theorem 4.9 proves a general multiplication-surjectivity statement for semistable vector bundles on an elliptic curve; Theorem 5.7 and Proposition 8.1 use this to prove degree-one generation for B(S^gE, sigma, L) and B(E^g, sigma, L_{n/k}); Theorem 6.9 and Theorem 9.7 then use Lemma 2.12 to reduce relation-degree generation to the same type of multiplication maps. None of these arguments assumes the surjectivity or relation-degree conclusions. The paper does import several structural facts from the authors' companion papers, notably the identification X_{n/k} with E^g or S^gE ([CKS19b, Cor. 4.24]), the vanishing of the quadratic relations on the graph of sigma' ([CKS19b, Cor. 5.9]), and some Hilbert-series and dimension computations ([CKS20]). These are load-bearing dependencies, but they are stated theorems from separate works with proofs that do not presuppose the present results. Such self-citation is not circular reasoning: no equation or construction in this paper reduces a predicted quantity to a fitted input, and no known empirical pattern is merely renamed. Even if a referee found a gap in the proof of Theorem 4.9, that would be a correctness defect, not a circularity defect. Accordingly, there is no circular step to record.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No parameters are fitted and no new entities are invented. The axioms are the standard toolkit of algebraic geometry and noncommutative algebra, plus several results imported from the authors' own companion papers. Those imports are not proved here and are the main non-standard dependencies.

assumptions (7)
  • standard math Standard framework of schemes, coherent sheaves, and graded algebras over C.
    Throughout the paper, including definitions of B(X,sigma,L) in Section 2.3 and Q_{n,k} in Section 1.4.
  • standard math Atiyah's classification of indecomposable vector bundles on an elliptic curve [Ati57].
    Used in Lemma 4.5 and Section 5.3 to assert indecomposable sheaves are semistable and to construct the bundle E_d.
  • standard math Maruyama's theorem that the tensor product of semistable sheaves over a curve is semistable in characteristic zero [Mar81].
    Invoked in Lemma 4.6 and repeatedly in Sections 5, 8, and 9.
  • standard math Artin-Van den Bergh equivalence QGr(B(X,sigma,L)) congruent to Qcoh(X) for sigma-ample L [AVdB90, Thms. 1.3 and 1.4].
    Used in Theorem 1.1(2), Corollary 2.8, and Section 2.3.3.
  • standard math Keeler's criterion: an ample invertible sheaf is sigma-ample iff the action of sigma* on NS(X)_C is quasi-unipotent [Kee00, Thms. 1.2 and 1.4].
    Used in Corollary 2.6 and Theorem 2.5.
  • domain assumption The definition and basic properties of Q_{n,k}(E,tau), including the characteristic variety X_{n/k} and the vanishing of the quadratic relations on the graph of sigma', are taken from the authors' companion papers [CKS18] and [CKS19b].
    Proposition 3.5 cites [CKS19b, Cor. 5.9]; Theorem 3.2 cites [CKS19b, Sections 3 and 4]. These are not proved in the present paper.
  • domain assumption The Hilbert-series result dim(Q_2) = (2k+1)(k+1) for Q_{2k+1,k}, cited as [CKS20, Thm. 5.10], is assumed.
    Used in Lemma 7.7 and Proposition 7.8.

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Pith. "Pith review of Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings." pith.science (2026). https://pith.science/paper/B3OGYVGK

@misc{pith2026190806525,
  author       = {Pith},
  title        = {Pith review of: Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B3OGYVGK}},
  note         = {Machine review of arXiv:1908.06525}
}
abstract

The elliptic algebras in the title are connected graded $\mathbb{C}$-algebras, denoted $Q_{n,k}(E,\tau)$, depending on a pair of relatively prime integers $n>k\ge 1$, an elliptic curve $E$, and a point $\tau\in E$. This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau)$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau)$. When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ we show the homomorphism $Q_{n,k}(E,\tau) \to B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ is surjective, that the relations for $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ are generated in degrees $\le 3$, and the non-commutative scheme $\mathrm{Proj}_{nc}(Q_{n,k}(E,\tau))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$, respectively. When $X_{n/k}=E^g$ and $\tau=0$, the results about $B(X_{n/k},\sigma',\mathcal{L}'_{n/k})$ show that the morphism $\Phi_{|\mathcal{L}_{n/k}|}:E^g \to \mathbb{P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

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