{"id":"148e0f43-1264-45a9-af99-9ec710a27689","arxiv_id":"1908.03139","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For regular cubic surfaces over any field, a closed point of degree coprime to 3 forces a closed point of degree 1, 4, or 10; for cubic 3-folds and 4-folds, symmetric products are stably birational.","lead":"This paper studies when a cubic surface or hypersurface has a closed point defined over a given field, assuming it has a zero-cycle of degree coprime to 3. It generalizes a classical result of Coray to non-perfect fields and proves new stable birational equivalences between symmetric products of cubic hypersurfaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 6.2 misidentifies the generic fiber: for cubic 4-folds the affine factor must have dimension 4, not 3, and the proof's F_{R,2n-1} should be F_{R,n+1}.","rationale":"The reader's designated weakest assumption is the Galois-orbit counting in Theorem 3.6, but their rationale also flags the wrong affine dimension in Proposition 6.2 and the index inconsistency in the generic fiber. I regard the Proposition 6.2 error as the more load-bearing issue because it makes a stated headline theorem dimensionally false as written and the proof cites propositions for the wrong varieties. The Galois-orbit step in Theorem 3.6 is underproved, but the counting argument can be completed: a Galois-invariant subset of a union of three conjugate rational curves has size divisible by 3 once nodal points are assigned to component orbits, so it is less likely to invalidate the descent. Since the Proposition 6.2 error is concrete but straightforwardly repairable, the appropriate disposition remains conditional acceptance after correction, matching the reader's verdict.","tokens_in":11469,"tokens_out":42270,"duration_ms":482386,"concrete_test":"Recompute the dimensions in Proposition 6.2 for n=4: dim Sym^8(X) = 8*4 = 32 and dim Sym^7(X) = 7*4 = 28, so any birational equivalence Sym^8(X) ~ Sym^7(X) x A^D requires D = 4. Then replace F_{R,2n-1} by F_{R,n+1} in the proof and verify that Proposition 5.4 applies to F_{R,5} (degree R = 7, projective dimension 5, which is odd) and that Proposition 5.3 applies to F_{R,4} (degree R = 5, projective dimension 4). If both checks pass, the stable birationality result stands after correcting the index and the dimension formula; if either fails, the proof of the main birationality claim is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 6.2 (n=3,4) claims that Sym^{n+4}(X) is birational to Sym^{2n-1}(X) times P^{15-3n}. The proof identifies the generic fiber of f_n: P -> (C_P ∩ X)\\P as F_{R,2n-1}, but this is not the variety that arises. Since X lies in P^{n+1}, C_P is a rational normal curve of degree n+1, so the family of such curves through a fixed universal zero-cycle R is F_{R,n+1}. For n=4, dim Sym^8(X) = 32 and dim Sym^7(X) = 28, so the complementary affine space must have dimension 4, whereas the printed formula gives 15-3n = 3. The cited applications of Proposition 5.3 to F_{R,5} and Proposition 5.4 to F_{R,7} are therefore applied to the wrong objects. With the corrected identifications F_{R,4} for n=3 and F_{R,5} for n=4, Proposition 5.3 (deg R = n+1) and Proposition 5.4 (deg R = n+2 with n odd) do apply, so the stable birationality conclusion appears repairable; however, the statement and proof as written are internally inconsistent and cannot be accepted without correction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies closed points and zero-cycles on cubic hypersurfaces over arbitrary fields. Its main results are: (i) Theorem 2.6, extending Coray's theorem to regular cubic surfaces over arbitrary fields by a lifting argument to mixed characteristic; (ii) Theorem 3.6, using rational normal curves to descend the degree of certain closed points on smooth cubic 3-folds and 4-folds; and (iii) Section 6, proving stable birationality statements for symmetric products, in particular Sym^7(X) and Sym^5(X) for cubic 3-folds and Sym^8(X) and Sym^7(X) for cubic 4-folds. The paper also develops auxiliary results on the rationality of moduli spaces F_{P,n} of rational normal curves through a fixed zero-cycle P.","tokens_in":11742,"tokens_out":38996,"duration_ms":418606,"significance":"If corrected, these are solid contributions. Theorem 2.6 is a genuine extension of Coray's theorem with a clean lifting-to-mixed-characteristic proof, and Theorem 3.6 offers a plausible new descent mechanism for cubic 3-folds and 4-folds. The symmetric-product results are interesting in the context of the Cassels--Swinnerton-Dyer circle of questions, and the explicit parametrizations in Proposition 5.4 are a useful addition. The paper relies on standard, well-documented machinery (Hilbert schemes, symmetric products, Galois descent, Rydh's cycle theory), and the main methods are transparent and reproducible from the text.","major_comments":[{"comment":"The displayed birationality has a dimension error for n=4, and the proof identifies the wrong generic fiber. For a smooth cubic n-fold X⊂P^{n+1}_k, a rational normal curve through a length-(n+4) point has degree n+1 in P^{n+1}, so the generic fiber of f_n is F_{R,n+1}, not F_{R,2n-1}. Consequently, for n=3 one should apply Proposition 5.3 to F_{R,4}, and for n=4 one should apply Proposition 5.4 to F_{R,5}. Also, dim Sym^8(X)=32 and dim Sym^7(X)=28, so the affine factor for n=4 must have dimension 4, whereas 15-3n gives 3; the correct dimension is n(5-n), namely 6 for n=3 and 4 for n=4. The stable birationality conclusion appears repairable with these corrections, but the statement and proof as printed are internally inconsistent.","section":"§6.2, Proposition 6.2"},{"comment":"The last paragraph of the proof asserts that a degree-3 component C2 which is a nodal union of three Galois-conjugate rational curves cannot contain the degree-8 point, because the number of geometric points would be a multiple of 3. This assertion is essential for the descent in the non-general-position case and is only sketched. A complete argument can be given: if P is a smooth geometric point on one component and H is the stabilizer (index 3) of that component, then the H-orbit of P has size [k(P):k]/[L∩k(P):k]=8, so all eight conjugates would lie on that component; applying a Galois element moving the component gives a further disjoint H-orbit, contradicting the total of eight points. For nodes, the Galois group permutes the three pairwise intersections transitively, so any Galois-invariant subset of nodes has cardinality divisible by 3. The paragraph should include this reasoning.","section":"§3.3, proof of Theorem 3.6"}],"minor_comments":[{"comment":"The Hilbert polynomial of a rational normal curve of degree n in P^n is h(t)=nt+1, not t+n+1; accordingly the Hilbert scheme should be Hilb^{nt+1,°}_{P^n_k/k}, and the curves in the definition of F_{P,n} have degree n, not n+1.","section":"§5.1, Proposition 5.1"},{"comment":"In the sentence 'If n or deg(P) is odd, then C_K is a conic with a closed point of odd degree', the word 'conic' should be replaced by 'curve of genus 0'; the logic is otherwise sound because, when n is odd, the line bundle O_C(1) has odd degree and forces C_K to be split.","section":"§5.2, proof of Proposition 5.2"},{"comment":"In the proof of Lemma 5.8, the coordinates {X_t} should be coordinates of P^{2d-1}_k, not P^{2d+1}_k, and the displayed sum defining f^* X_t should run to 2d-1 rather than 2d+1.","section":"§5.4, Lemma 5.8"},{"comment":"The subscript in F_{P,n} denotes the ambient projective dimension, but in the proof of Proposition 6.2 the expressions F_{R,5} and F_{R,7} are used as though the subscript were a degree; after correcting the generic fiber to F_{R,n+1}, this notation should be made explicit to avoid confusion.","section":"§6.2, Proposition 6.2"},{"comment":"The application of Lemma 2.5 to the relative Hilbert scheme Hilb^d_{X/R} over R deserves a brief justification: since X/R is flat and the special fiber is regular at P, the point [P] is a smooth point of the relative Hilbert scheme over R, so Hensel lifting applies.","section":"§2.2, proof of Theorem 2.6"}],"recommendation":"major_revision","confidential_remarks":"The paper's main ideas are sound and the errors are localized; in particular, the problem in Proposition 6.2 is a misidentification of the generic fiber and a wrong affine-space dimension, both of which are repairable without changing the stable birationality claims. I see no concerns about originality or citation practices."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take on Ma's \"Closed points on cubic hypersurfaces.\" The paper is worth reading and refereeing, but Section 6.2 as written is internally inconsistent.\n\nThe main result, Theorem 2.6, removes the perfectness assumption in Coray's theorem for regular cubic surfaces. The lifting-to-characteristic-0 trick is genuinely new, and the proof is convincing, modulo a small point about lifting the Hilbert point. The use of Lemma 2.3 to extract a closed point of degree prime to 3 is fine. I think that theorem is correct.\n\nThe stable birationality results for symmetric products of cubic 3-folds and 4-folds are new and interesting. The strategy is nice: dominant maps between symmetric products via intersection with rational normal curves, then study the rationality of the generic fiber. The generic fiber is the moduli space of rational normal curves through a fixed zero-cycle, denoted F_{P,n}. Propositions 5.3 and 5.4 give rationality in the needed degrees.\n\nBut Proposition 6.2 has a real error. For X in P^{n+1}, rational normal curves have degree n+1, not degree 2n-1. The generic fiber of f_n should be F_{R,n+1}, not F_{R,2n-1}. For n=3 that means F_{R,4}; for n=4, F_{R,5}. The printed proof applies Proposition 5.3 to F_{R,5} and Proposition 5.4 to F_{R,7}, which are the wrong objects. With the corrected indexing the cited propositions do apply: Proposition 5.3 (deg = n+1) for the first, Proposition 5.4 (deg = n+2, odd ambient dimension) for the second. Also the affine factor dimension is wrong: the complement dimension for n=4 is 4, not 3, so it should be P^4, not P^{15-3n} (which gives P^3). The conclusion is salvageable, but the statement and proof need rewriting.\n\nThe other weak spot is Theorem 3.6. The argument that a degree-3 component cannot be a nodal union of three Galois-conjugate rational curves is asserted in a few lines and not proved. The orbit counting is plausible, but it is essential for the descent, so it needs a real proof.\n\nThere are also several typos: Proposition 5.1 writes P^3 instead of P^n in the Hilbert scheme, and Proposition 5.2 calls the curve a conic. None of these affect the main ideas.\n\nProposition 5.4 overlaps with Florence–Reichstein (as noted in Remark 5.8.1), but the proof here is independent and explicit. The citation handling is honest.\n\nIn sum: a solid contribution to arithmetic and birational geometry, with one clear flaw in a key proposition that is repairable and one sketchy argument that needs expansion. It deserves a serious referee, and I'd send it to one, asking for a revision.\n\nBest,\n\n[your name]","headline":"A genuinely new generalization of Coray's theorem plus appealing stable birationality results, but Proposition 6.2 has a real indexing/dimension error that needs fixing before the paper is accepted.","tokens_in":12314,"tokens_out":5087,"would_cite":false,"duration_ms":47969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J70","14E08","14G05","14C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A closed point whose degree is prime to 3 on a regular cubic surface forces a closed point of degree 1, 4, or 10, even over imperfect fields.","keywords":["closed points","cubic surfaces","cubic threefolds","cubic fourfolds","rational normal curves","symmetric products","stable birationality","imperfect fields"],"falsifier":"Find a smooth cubic fourfold over a field with a degree-8 closed point such that the specialized rational normal curve has a degree-3 component that is a nodal union of three Galois-conjugate rational curves and the residual intersection produces no point of degree 1, 2, 4, 5, or 7; equivalently, exhibit eight Galois-conjugate points on such a curve whose orbit sizes are not all multiples of 3.","tokens_in":11234,"feed_emoji":"","tokens_out":10495,"duration_ms":100313,"temperature":0.7,"pith_summary":"This paper proves that on a regular cubic surface over any field, a closed point whose residue-field degree is not divisible by 3 forces a closed point of degree 1, 4, or 10. That removes the perfect-field assumption from a theorem known before only in that setting, and it does so by lifting the surface to a discrete valuation ring and using smoothness of the Hilbert scheme to descend degree steps. For cubic threefolds and fourfolds, the paper shows that degree-7 and degree-8 closed points can be lowered to points of degree at most 5 or 7 by cutting the hypersurface with the unique rational normal curve through the geometric points of the original point. The same construction yields birational equivalences between symmetric products: for a smooth cubic threefold, $\\operatorname{Sym}^7(X)$ and $\\operatorname{Sym}^5(X)$ are stably birational, and for a smooth cubic fourfold, $\\operatorname{Sym}^8(X)$ and $\\operatorname{Sym}^7(X)$ are.","feed_headline":"A degree prime to 3 point forces degree 1, 4, or 10 on cubic surfaces","feed_subtitle":"Generalizes the perfect-field theorem and shows Sym^7 and Sym^5 of a cubic threefold are stably birational.","key_machinery":"The load-bearing object is the rational normal curve through a closed point of degree $n+3$ in $\\mathbb{P}^n$. When the geometric points are in linearly general position, a Cremona transformation centered at a subset of them identifies such curves with lines avoiding certain planes, which proves there is exactly one such curve. When the geometric points are not in linear general position, the paper perturbs the point to general position over the generic point, obtains the unique curve there, and specializes it; the special fiber is a reduced curve of degree at most 5, and its intersection with the cubic produces a residual zero-cycle of lower degree. For the birationality results, the same curve is used to define a rational map $\\operatorname{Sym}^{n+4}(X) \\to \\operatorname{Sym}^{2n-1}(X)$ by sending a subscheme $P$ to the residual intersection $(C_P \\cap X) \\setminus P$; the rationality of the generic fiber is analyzed through the varieties $F_{P,n}$ parameterizing genus-zero curves through a fixed subscheme.","core_discovery":"On its own terms, the central claim is Theorem 2.6: if a regular cubic surface over a field $k$ contains a closed point $P$ whose degree is prime to 3, then it contains a closed point of degree 1, 4, or 10. The proof keeps the overall shape of the perfect-field descent but replaces separability arguments with a lift to a complete discrete valuation ring with residue field $k$; the point $P$ lifts to the generic fiber, the known theorem over perfect fields applies there, and the resulting lower-degree point specializes back. The paper's second main claim is Proposition 6.2: for a smooth cubic $n$-fold with $n=3$ or $4$, the symmetric product $\\operatorname{Sym}^{n+4}(X)$ is birational to $\\operatorname{Sym}^{2n-1}(X)$ times an affine space, so the two symmetric powers are stably birational.","pith_inferences":["The parity condition in the stable-birationality statements may be an artifact of the rationality proof for $F_{P,n}$; if the even-even case of Question 5.9 has a positive answer, the birationality would hold without the parity restriction.","The same residual-intersection construction should produce stable birationalities for smooth hypersurfaces of degree $m$, along the lines of the paper's remark that $\\operatorname{Sym}^l(X)$ and $\\operatorname{Sym}^{m(n+3)-l}(X)$ are stably birational when $l$ or $n$ is odd; removing the parity assumption is a natural next step.","The unproved orbit-counting assertion could be checked exhaustively by computer for small fields: list the possible Galois orbit structures of eight points on a degree-3 nodal rational curve; any orbit structure not consisting of multiples of 3 would show the current justification needs repair, even if the theorem remains true."],"forward_implications":["On a regular cubic surface over any field, the question of whether a point of degree prime to 3 exists is settled exactly by looking for points of degrees 1, 4, and 10.","Smooth cubic threefolds with a degree-7 point and smooth cubic fourfolds with a degree-8 point automatically contain a point of one of the listed lower degrees.","The symmetric powers $\\operatorname{Sym}^7(X)$ and $\\operatorname{Sym}^5(X)$ are stably birational for a cubic threefold, and $\\operatorname{Sym}^8(X)$ and $\\operatorname{Sym}^7(X)$ for a cubic fourfold.","The rational map from $\\operatorname{Sym}^d(X)$ to $\\operatorname{Sym}^e(X)$ turns a length-$d$ subscheme of $X$ into a $k$-point of $\\operatorname{Sym}^e(X)$, so the existence statements for closed points can be read as statements about points on symmetric powers."],"supporting_citations":[{"why":"Supplies the perfect-field theorem and the descent argument for cubic surfaces that the paper extends to arbitrary fields.","marker":"[Cor76]"},{"why":"Provides the Hilbert scheme smoothness, DVR lifting, and valuative criteria used throughout the lifting argument.","marker":"[Sta19]"},{"why":"Gives the specialization map property for Henselian local rings used to lift the closed point to characteristic zero.","marker":"[Gro67]"},{"why":"Shows symmetric powers of projective space are rational, used to prove stable rationality of $F_{P,n}$.","marker":"[Mat68]"},{"why":"Provides the lemma that a rational map to a proper variety with a smooth point yields a point, used in Proposition 4.3.","marker":"[Poo17]"},{"why":"Supplies the Hilbert–Chow morphism used to pass from Hilbert schemes to symmetric products.","marker":"[Nak99]"},{"why":"Used to analyze $k$-points on symmetric products in positive characteristic and bound the resulting zero-cycle degree.","marker":"[Ryd08]"}],"fun_headline_variants":["Degree prime to 3 point on cubic surface yields degree 1, 4, or 10","Prime-to-3 degree point on cubic surface forces degree 1, 4, or 10","Cubic threefold symmetric powers Sym^7 and Sym^5 are stably birational","Closed points on cubic hypersurfaces: symmetric products stably birational","Cubic surface point of prime-to-3 degree forces low-degree point"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The descent for degree-8 points on cubic fourfolds depends on an unproved orbit-counting step: if the specialized curve has a degree-3 component that is a nodal union of three Galois-conjugate rational curves, the proof assumes the eight geometric points must form orbits whose sizes are multiples of 3, and this assertion is made without a full proof.","fun_headline_variants_meta":{"raw":{"variants":["Degree prime to 3 point on cubic surface yields degree 1, 4, or 10","Prime-to-3 degree point on cubic surface forces degree 1, 4, or 10","Cubic threefold symmetric powers Sym^7 and Sym^5 are stably birational","Closed points on cubic hypersurfaces: symmetric products stably birational","Cubic surface point of prime-to-3 degree forces low-degree point"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000575,"raw_usage":{"total_tokens":2615,"prompt_tokens":747,"completion_tokens":1868,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":363,"completion_tokens_details":{"reasoning_tokens":1766}},"tokens_in":363,"tokens_out":1868,"duration_ms":13403,"temperature":1.0,"reasoning_tokens":1766,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:31:15.094134+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a smooth cubic fourfold over a field with a degree-8 closed point such that the specialized rational normal curve has a degree-3 component that is a nodal union of three Galois-conjugate rational curves and the residual intersection produces no point of degree 1, 2, 4, 5, or 7; equivalently, exhibit eight Galois-conjugate points on such a curve whose orbit sizes are not all multiples of 3.","supporting_citations":[],"review_version":1}