{"id":"297f37d0-43b5-46b1-a10c-7e99afb403de","arxiv_id":"1908.06525","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Feigin-Odesskii elliptic algebras whose characteristic variety is a product or symmetric product of an elliptic curve, the canonical map to the twisted homogeneous coordinate ring is surjective and its relations are generated in degrees at most three.","lead":"This paper shows that a family of noncommutative algebras built from elliptic curves, the Feigin-Odesskii algebras, map onto the twisted coordinate rings of their characteristic varieties in the two main geometric cases. It also proves that these varieties appear as closed subschemes of the algebras' noncommutative projective spaces, yielding classical results about how powers of an elliptic curve embed in projective space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.9 proof contains an invalid identification, weakening the surjectivity results that underpin the main claims.","rationale":"The reader correctly identified the dependence on companion-paper results (Proposition 3.5 and the identification of X_{n/k}) as a condition for accepting the main theorem. However, the more immediate load-bearing concern is internal: the proof of Theorem 4.9, which is developed in this paper and used heavily in Sections 5–9, contains an invalid step. The diagram in Section 4.2 does not establish the claimed isomorphism between the kernel of H^0(K^∨)⊗O→K^∨ and U^∨; indeed, U^∨ has no global sections when deg(U)>0, so the two sheaves cannot generally be identified. Since Theorem 4.9 and its corollaries feed directly into the proofs of the surjectivity and relation-degree statements in Theorem 1.1, this proof gap is a significant correctness risk. That said, the theorem's statement is plausible and likely true, and the paper contains substantial independent value in its detailed reductions to semistable bundles. The external dependencies flagged by the reader are real but are a normal feature of a research series; they can be checked in the cited preprints. The proper response is to keep the conditional verdict, broadening the conditions to include a rigorous proof of Theorem 4.9. I therefore leave the verdict unchanged.","tokens_in":50930,"tokens_out":25833,"duration_ms":218574,"concrete_test":"Re-derive the indecomposability step in Theorem 4.9 directly: take an elliptic curve E and U = O(3) (or another indecomposable semistable bundle with µ(U)>2), compute K = ker(H^0(U)⊗O_E→U), and compute the kernel M of H^0(K^∨)⊗O_E→K^∨. Verify whether M is isomorphic to U^∨. If M is not isomorphic to U^∨, the proof's central inference fails. Then check whether M is nonetheless indecomposable for this example and for a range of U, V; if indecomposability cannot be established, attempt to prove Theorem 4.9 using the filtration in Lemma 4.12 or another argument. A successful independent proof of Theorem 4.9 (or a replacement for Corollary 4.11) would settle whether the gap is merely expository or substantive.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4.9 (Section 4.2) aims to show that K = ker(H^0(E,U)⊗O_E → U) is indecomposable by proving that the kernel M of the evaluation map H^0(E,K^∨)⊗O_E → K^∨ is indecomposable. The argument claims M is isomorphic to U^∨, hence indecomposable because U is indecomposable. However, the commutative diagram in the proof does not identify M with U^∨. Chasing the diagram gives only a monomorphism H^0(U^∨)⊗O_E ↪ U^∨, and because deg(U^∨)<0 (U is generated by global sections and semistable, so deg U>0), Lemma 4.8(2) implies H^0(U^∨)=0. Thus the claimed isomorphism cannot hold in general; the proof supplies no valid reason for M to be indecomposable. This matters because Theorem 4.9 is the engine behind Corollaries 4.10 and 4.11, Proposition 5.6, Lemma 6.8, and ultimately Theorems 5.7, 6.9, 8.1, and 9.7, which are precisely the results used to establish Theorem 1.1(4) and (5). If Theorem 4.9 is not proven, the paper's central surjectivity and relation-degree results are unsupported by the arguments given, even if the statement itself is true and could be proved by other means.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":51275,"tokens_out":45630,"duration_ms":364679,"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":[{"comment":"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.","section":"§4.2, proof of Theorem 4.9"},{"comment":"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.","section":"Theorem 1.1(3)–(5); §7.3 and §7.4"},{"comment":"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.","section":"§3.1.7 and Corollary 3.6 (Prop. 3.5)"}],"minor_comments":[{"comment":"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.","section":"§7.4, Lemma 7.7"},{"comment":"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.","section":"§4.2"},{"comment":"Two lines before Corollary 3.6, \"Qn/k(E,τ)\" should read \"Q_{n,k}(E,τ)\".","section":"§3.2"},{"comment":"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.","section":"§9.2"}],"recommendation":"major_revision","confidential_remarks":"The reader's conditional verdict and the skeptic's note were taken into account. On the substantive question: the identification M ≅ U^∨ in Theorem 4.9 is actually valid once the Serre-duality step is corrected; the skeptic's claim that the identification \"cannot hold in general\" is not sustained. The real problems are (i) the false intermediate statement \"H^1(ε) is an isomorphism\" in the proof of Theorem 4.9, and (ii) the missing [2^{g−1},m] case in the proof of Theorem 1.1(4)–(5). Both are local and repairable. I recommend verifying [CKS19b, Cor. 5.9] and [CKS20, Thm. 5.10] directly, since the paper's central conclusions collapse if those fail. The paper is otherwise careful, and its geometric corollary (τ=0: projective normality and quadric–cubic generation for |L_{n/k}|) is a notable concrete payoff."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me give you the short version: this is a serious paper that likely proves the right things, but there is a hole in the proof of Theorem 4.9, and since that theorem is load-bearing for the surjectivity results, the authors need to fix it or spell out the derived-category argument.\n\nWhat is actually new: for k>1, when the characteristic variety X_{n/k} is E^g or S^gE, the canonical map Ψ from Q_{n,k}(E,τ) to the twisted coordinate ring is surjective, the relations are generated in degrees 2 and 3, and you get the projective normality consequences. The reduction of these questions to multiplication maps for semistable sheaves on an elliptic curve is clean, and the treatment of the S^gE case as a projective bundle over E, with explicit NS classes and the description of σ′ as a translation, is genuinely useful. The special case Q_{2k+1,k} and the 5 cubic generators for Q_{5,2} are nice concrete payoffs.\n\nThe soft spot is real. In the proof of Theorem 4.9, the step showing K is indecomposable uses a diagram where H^0(ε^∨) is claimed to be an isomorphism by Serre duality. As written that doesn't hold: H^1(H^0(U)⊗O) is nonzero on an elliptic curve, so the displayed exact sequence and the Serre duality step don't line up. Chasing the diagram gives at most a monomorphism from H^0(U^∨)⊗O into U^∨, and since H^0(U^∨)=0 that doesn't identify the kernel with U^∨. So the indecomposability of K is not established by that argument. This matters because Theorem 4.9 feeds directly into the multiplication results in Sections 5, 6, and 9.\n\nHowever, the authors themselves sketch an alternative proof in Remark 4.3.3 using the Seidel–Thomas twist functor, which should work: O is spherical on an elliptic curve, the twist is an autoequivalence, and K[1] is the twist of U. That argument is not written out, but it is credible. So I would not bet against the theorem. Still, a referee should insist on a clean proof.\n\nAlso be aware the dependence on companion papers is heavy: the existence of Ψ relies on Proposition 3.5 from [CKS19b], and the identification X_{n/k}≅E^g or S^gE is also cited. That's normal for a series, but it means the paper is not self-contained.\n\nBottom line: send it to peer review. The main theorems are important and the overall architecture is sound. The authors should be asked to fix Theorem 4.9 or make the derived-category proof precise.","headline":"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.","tokens_in":51809,"tokens_out":11924,"would_cite":true,"duration_ms":103830,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14A22","16S38","16W50","14H52","14F05"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["elliptic algebras","Feigin–Odesskii algebras","twisted homogeneous coordinate rings","characteristic variety","elliptic curve","symmetric power","projective normality","semistable vector bundles"],"falsifier":"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.","tokens_in":50762,"feed_emoji":"🔗","tokens_out":9827,"duration_ms":85032,"temperature":0.7,"pith_summary":"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.","feed_headline":"Elliptic algebras map onto their characteristic-variety rings","feed_subtitle":"For characteristic varieties E^g and S^gE the map is onto, and the relations are generated in degrees 2 and 3.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $X_{n/k}$."],"forward_implications":["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 $aD+bF$, $a\\ge1$, $b\\ge2$."],"supporting_citations":[{"why":"Supplies the vanishing of $Q_{n,k}$'s quadratic relations on the graph of $\\sigma'$, the fact that makes the canonical homomorphism exist.","marker":"[CKS19b, Cor. 5.9]"},{"why":"Establishes the basic properties of the sheaf $L_{n/k}$: base-point free, ample, very ample when all $n_i\\ge3$, with $n$-dimensional space of sections.","marker":"[CKS19b, §§3 and 4]"},{"why":"Identifies $X_{n/k}$ with $E^g$ or $S^gE$ under the continued-fraction hypotheses used in Theorem 1.1(3).","marker":"[CKS19b, Cor. 4.24]"},{"why":"Provides the equivalence $QGr(B(X,\\sigma,L)) \\simeq Qcoh(X)$ for $\\sigma$-ample $L$, which converts algebraic statements about $B$ into geometric statements about $X$.","marker":"[AVdB90, Thms. 1.3 and 1.4]"},{"why":"Gives criteria for $\\sigma$-ampleness, used to know that $B(E^g,\\sigma,L)$ has good quotient-category behaviour for translation automorphisms.","marker":"[Kee00, Thms. 1.2 and 1.4]"},{"why":"The comparison of $B(X_{n/k},\\sigma',L')$ with the invariant subring $B(E^g,\\sigma,L)^{\\Sigma}$ follows the pattern of this earlier result.","marker":"[ST94, Prop. 2.5]"}],"fun_headline_variants":["Elliptic algebras surject onto twisted rings for E^g and S^gE","Surjective map to twisted rings, relations in degrees ≤3","Quadrics and cubics cut out projectively normal E^g","Closed E^g or S^gE in noncommutative Proj","Elliptic algebra map yields closed subvariety E^g or S^gE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Elliptic algebras surject onto twisted rings for E^g and S^gE","Surjective map to twisted rings, relations in degrees ≤3","Quadrics and cubics cut out projectively normal E^g","Closed E^g or S^gE in noncommutative Proj","Elliptic algebra map yields closed subvariety E^g or S^gE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000474,"raw_usage":{"total_tokens":2467,"prompt_tokens":1174,"completion_tokens":1293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":790,"completion_tokens_details":{"reasoning_tokens":1192}},"tokens_in":790,"tokens_out":1293,"duration_ms":11234,"temperature":1.0,"reasoning_tokens":1192,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:42:37.949901+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}