{"id":"4590a83d-cb3c-4a33-a9c7-7ae082c6643e","arxiv_id":"1908.05449","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper constructs closed embeddings of products of Grassmannians into Grassmannians of tensor, exterior, and symmetric powers of the underlying vector bundles, and determines the precise characteristic condition under which the symmetric-power embedding is closed.","lead":"This paper proves that several natural ways of mapping Grassmannian varieties into larger ones, by taking tensor, exterior, or symmetric powers of the underlying vector spaces, are closed embeddings, generalizing the classical Segre and Plücker embeddings. It also pins down exactly when the symmetric-power embedding works, including a surprising failure in characteristic p.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.7's stated hypothesis 'Z[rm−2]-algebra' is not merely obscure: read literally as Z[1/(rm−2)], it makes the lemma false, and Prop. 3.7/3.14 inherit this. The intended condition is evidently 'm≥2 or r invertible,' but the written proof of Theorem 3.15 for S_r needs that correction.","rationale":"The central claim of Theorem 3.15 is otherwise plausible and well-supported: T, Tr, and Ar are proper monomorphisms, and the closed immersion argument via EGA is standard. The only delicate point is the symmetric-power monomorphism Sr, which rests entirely on Lemma 2.7 / Prop. 3.7 / Prop. 3.14. The reader noticed the same lemma and flagged the 'Z[rm−2]' phrase as obscure; my pass shows it is worse than obscure. Under the standard reading as Z[1/(rm−2)], Prop. 3.14 is false, and the author's own Remark 3.16 supplies the counterexample (m=1, r=3 over F_3[ε]/(ε^3)). Thus the written proof has a formal gap: Theorem 3.15's Sr half cites a proposition whose stated hypothesis is not the one used in the proof. The fix is a one-line correction of the hypothesis, so I would not reject the paper's mathematics; I would make acceptance conditional on that correction and on expanding the terse 'by symmetry' step in Lemma 2.7, which is repairable by permuting the basis of V2. No independent evidence such as a machine-checked proof or reproducible code is present, so the formal statement matters.","tokens_in":9513,"tokens_out":28163,"duration_ms":274950,"concrete_test":"The decisive check is to apply the literal statement of Lemma 2.7 to R=F_3[ε]/(ε^3), m=1, r=3, W=Re1⊕Re2, V1=Re1, V2=R(e1+εe2). Since R is a Z[1/(rm−2)] = Z[1]-algebra, the written hypothesis is satisfied but the conclusion V1=V2 is false, so the statement as printed must be changed. Then verify that after replacing 'Z[rm−2]-algebra/scheme' by 'm≥2 or r invertible', the coefficient computation in (2.9)–(2.11) goes through for r=2,3 with no use of r-invertibility in the m≥2 case, e.g. by checking the matrix identity Sym^{r−1} bar A · nu = 0 in a computer algebra system.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 2.7 states: let R be a Z[rm−2]-algebra. The author glosses this as 'm≥2 or r invertible', but the notation does not say that. If 'Z[rm−2]' is read as Z[1/(rm−2)], the lemma is false: take m=1, r=3, R=F_3[ε]/(ε^3), W=R e1⊕R e2, V1=R e1, V2=R(e1+ε e2). Then Sym^3 V1 and Sym^3 V2 are both R e1^3 (because (e1+ε e2)^3=e1^3 in char 3), but V1≠V2. This is exactly the counterexample class the author himself gives in Remark 3.16 for r not invertible. The proof of Lemma 2.7 only uses the dichotomy m≥2 (coefficient argument, no r-invertibility) and, for m=1, r∈R^×; the stated 'Z[rm−2]' condition is therefore the wrong formal hypothesis. Prop. 3.7 and Prop. 3.14 use the same phrase, so the proof of Theorem 3.15's S_r half is not valid as written. The intended repair is clear: replace the hypothesis by 'm≥2 or r is invertible on X', and handle r=1 trivially. This is a statement-level bug, not a counterexample to the theorem, but it must be corrected before the S_r claim is established by the given proof.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs natural X-morphisms between Grassmannian varieties attached to a scheme X: the tensor-product morphism T, the tensor-power morphism Tr, the exterior-power morphism Ar, and the symmetric-power morphism Sr. The main theorem (Theorem 3.15) asserts that T, Tr, and Ar are closed immersions and that Sr is a closed immersion provided m>1 or r is invertible on X. The proof has two parts: purely multilinear lemmas (Lemmas 2.5–2.7) showing that certain tensor, exterior, and symmetric powers of inclusions determine the inclusions, sheafified in Propositions 3.1–3.7; and a geometric argument (Theorem 3.15) that combines monomorphicity with properness and finite presentation, following EGA, to conclude closed immersion. The paper also includes a remark (3.16) showing that the condition on Sr is necessary.","tokens_in":9902,"tokens_out":23371,"duration_ms":231865,"significance":"If the main theorem is established, the paper provides a useful, functorial family of closed embeddings generalizing Segre and Plücker embeddings, with potential applications in Schubert calculus, intersection theory, and the author's work on Rapoport-Zink towers. The proof strategy is clean: the local multilinear lemmas are stated in enough generality, and the use of the EGA proper-monomorphism criterion avoids explicit equations. The author also correctly identifies (Remark 3.16) the exact obstruction for the symmetric-power map. However, the formal hypothesis in Lemma 2.7 and its sheafified versions is incorrect as written and must be repaired before the Sr part of the theorem is proved. The fix appears straightforward and does not affect the geometric strategy.","major_comments":[{"comment":"The hypothesis 'R is a Z[rm−2]-algebra' is not equivalent to 'm≥2 or r is invertible in R' as claimed in the proof. If the phrase is read literally as adjoining the integer rm−2 to Z, then Z[rm−2]=Z and every ring satisfies the hypothesis, which is false. If it is read as the common localization Z[1/(rm−2)], it is still not equivalent. The lemma is false under the stated hypothesis: take m=1, r=3, R=F_3[ε]/(ε^3), W=R e1⊕R e2, V1=R e1, V2=R(e1+ε e2); then Sym^3 V1=Sym^3 V2=R e1^3 in Sym^3 W, but V1≠V2. This is precisely the phenomenon described in Remark 3.16. Consequently Proposition 3.7 and Proposition 3.14, and hence the proof of the Sr half of Theorem 3.15, are not valid as written; for example, when X=Spec(F_2), m=2, r=3, the intended condition m>1 holds, but F_2 is not a Z[1/4]-algebra, so the stated hypothesis fails. The intended hypothesis is clearly 'm≥2 or r is invertible', and the proof of Lemma 2.7 actually proves the lemma under that condition; the formal statements must be corrected accordingly.","section":"Lemma 2.7, Props 3.7 and 3.14"},{"comment":"The step 'This implies the same statement for all r+1-minors' is not justified as written. If an (r+1)-minor contains exactly one of the last n−m rows, expansion along that row produces r-minors that contain no last row, so the previously established vanishing does not apply to them. The desired conclusion is nevertheless true: fixing a last row c, the vanishing of all r-minors involving c and any r−1 rows of the invertible top block forces c to be zero, because the top rows form a basis. This argument should be included; without it, the proof of Lemma 2.6, and hence of the Ar part of Theorem 3.15, is incomplete as written.","section":"Lemma 2.6"}],"minor_comments":[{"comment":"There are several typos: 'short exact sentences' should be 'short exact sequences', 'for 1≤r≤r' should be 'for 1≤r≤n', and in Proposition 2.1 the phrase 'for every 1≤d≤n' applies only to parts (2) and (3), not to part (1).","section":"Introduction and Section 2"},{"comment":"The proof refers to diagrams '(3.5)' and '(3.6)', but the factorization diagrams in Proposition 3.1 are numbered (3.2) and (3.3); the cross-references should be corrected.","section":"Proposition 3.1, proof"},{"comment":"The reduction 'By symmetry, we can assume j=m+1 and i=1' is correct but terse: it uses permutation of the basis vectors e_{m+1},...,e_n and a change of basis of V2. A brief justification would improve readability.","section":"Lemma 2.7"},{"comment":"The sentence 'The other cases are similar' is very brief; for Tr one should explicitly cite the composition T∘Δ_r and the closedness of the diagonal, and for Sr one should invoke the corrected Proposition 3.7 with the intended hypothesis.","section":"Theorem 3.15, proof"},{"comment":"The existence of a local Noetherian ring R with an element α of square zero and a morphism Spec(R)→X can be made explicit by taking R=κ(x)[ε]/(ε^2) for a point x of X whose residue field has characteristic p dividing r; the current wording is somewhat compressed.","section":"Remark 3.16"}],"recommendation":"major_revision","confidential_remarks":"The formal bug in Lemma 2.7 is real but appears to be a statement-level error rather than a counterexample to the intended theorem; the counterexample in the report is exactly the phenomenon the author discusses in Remark 3.16. I would invite a revision rather than reject. The paper is short and the fix is local, but it is load-bearing for the Sr half of the main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves that the natural tensor-product, tensor-power, and exterior-power maps between Grassmannian bundles are closed immersions, and that the symmetric-power map is a closed immersion when m > 1 or r is invertible. That is a clean, useful result, and the proof strategy is sound: check monomorphy via multilinear algebra, then use properness and finite presentation to upgrade to a closed immersion. The linear algebra lemmas for the tensor and exterior cases are written out in detail, and the characteristic-p caveat for symmetric powers is correct and interesting, with a genuine counterexample when m = 1 and r is not invertible.\n\nThe main problem is Lemma 2.7. The hypothesis says R is a \"Z[rm-2]-algebra,\" and the proof then asserts this means \"m ≥ 2 or r is invertible in R.\" That is not what the notation says. If read literally as Z[1/(rm-2)], the lemma is false: for m = 1, r = 3, R = F_3[ε]/(ε^3), the submodules spanned by e1 and by e1 + εe2 have the same third symmetric power but are distinct. The author's own Remark 3.16 gives exactly this counterexample for r not invertible. So the intended hypothesis is clearly \"m ≥ 2 or r ∈ R^×,\" and the proof works under that hypothesis, but as written the statement and the proofs of Proposition 3.7 and Proposition 3.14 inherit a bad formal condition. This is a statement-level bug, not a counterexample to the theorem, but it must be fixed before the S_r half is established by the given argument.\n\nTwo smaller issues: Propositions 3.4 and 3.7 say \"proof is similar\" and omit the argument, which is acceptable but annoying for a paper whose main selling point is self-containedness. The \"by symmetry\" step in Lemma 2.7 is also terse; a referee should ask for one or two more lines there. These are minor and do not affect my confidence in the tensor and exterior cases.\n\nWho is this for? Algebraic geometers working with Grassmannians, Schubert calculus, or Rapoport-Zink towers. The paper is short, readable, and the main result deserves to be in the literature after a routine revision.\n\nRecommendation: send to peer review. The referee should insist on a corrected Lemma 2.7 and expanded proofs for the two omitted propositions, but the central theorem is solid and the paper is worth publishing.","headline":"The main theorem is a useful and probably correct generalization of the Segre and Plücker embeddings, but Lemma 2.7's stated hypothesis is wrong and the symmetric-power half of the proof is not valid as written.","tokens_in":10372,"tokens_out":2441,"would_cite":true,"duration_ms":24731,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that tensor-product, tensor-power, and exterior-power maps between Grassmannian schemes are closed immersions, and that the symmetric-power map is a closed immersion exactly when the subbundle rank is at least 2 or r is…","keywords":["Grassmannian varieties","closed immersions","Segre embedding","Plücker embedding","symmetric powers","exterior powers","tensor products of vector bundles","flag varieties"],"falsifier":"Work out Lemma 2.7 for m=2 and r=2 over a local ring of characteristic 2: write two rank-2 subbundles whose $Sym^{2}$ images in $Sym^{2}$ W agree while the subbundles differ by a nilpotent matrix entry; if such a pair exists, Sr is not a closed immersion, contradicting Theorem 3.15, while if no such pair exists, the coefficient argument is sound.","tokens_in":9338,"feed_emoji":"📐","tokens_out":7931,"duration_ms":77611,"temperature":0.7,"pith_summary":"This paper proves that natural maps between Grassmannian varieties built from tensor operations on vector bundles are closed immersions. Specifically, the tensor-product map T, the tensor-power map Tr, and the exterior-power map Ar are closed embeddings, and the symmetric-power map Sr is a closed embedding whenever the subbundle rank m is at least 2 or r is invertible on the base scheme. These maps generalize the classical Segre and Plücker embeddings to arbitrary Grassmannians over any scheme. A sympathetic reader would care because closed embeddings of this kind give concrete geometric realizations of products and power constructions on subbundles, useful in intersection theory, Schubert calculus, and related enumerative questions.","feed_headline":"Tensor, wedge, and symmetric powers give closed Grassmannian embeddings","feed_subtitle":"Generalizing Segre and Plücker, the symmetric-power case needs rank at least 2 or an invertible r.","key_machinery":"The driving object is the functor of points of the Grassmannian, on which the morphisms are defined by tensoring, exterior-powering, or symmetric-powering the universal short exact sequence 0→F→V_S→G→0. The load-bearing identities are the determinant formulas det(M_1⊗...⊗M_r)=∏ det(M_i)^{...}, det(Sym^d M)=det(M)^{...}, and det(⋀^d M)=det(M)^{...} from Proposition 2.1, together with the recovery lemmas that an isomorphism between the tensor, exterior, or symmetric power of two subbundles forces an isomorphism of the subbundles themselves. These are sheafified and then combined with the criterion that a proper, finite-presentation monomorphism between schemes is a closed immersion.","core_discovery":"The central claim is Theorem 3.15: for finite locally free sheaves V_i and V on a scheme X, the morphisms T, Tr, and Ar between Grassmannian schemes are closed immersions, and Sr is a closed immersion if m>1 or r is invertible on X. The proof reduces the global statement to local multilinear algebra: Lemmas 2.5, 2.6, and 2.7 show that the tensor, exterior, and symmetric powers of an injective morphism of projective modules determine the original submodule, under the stated hypotheses. Sheafified as Propositions 3.1, 3.4, and 3.7, this gives monomorphisms on functors of points; because the Grassmannians involved are proper and of finite presentation over X, properness plus monomorphism forces closed immersion via a standard criterion.","pith_inferences":["A testable extension is to formulate the flag-variety analogues explicitly: applying tensor or exterior operations to a full flag should give closed embeddings of flag varieties under rank-compatibility conditions, following the paper's induction suggestion.","The coefficient computation in Lemma 2.7 suggests that Sym^r is blind to nilpotent line subbundles when r is divisible by the characteristic; the same phenomenon might be probed for m≥2 over mixed-characteristic bases where r is not invertible, which would mark the exact boundary of the theorem.","Because the recovery lemmas state that a subbundle is determined by its tensor, exterior, or symmetric power as a subobject, the embeddings give a way to compare moduli problems: a map from a scheme S to the target Grassmannian that lands in the image corresponds uniquely to a collection of subbundles with prescribed tensor relations.","The determinant identities in Proposition 2.1 remain valid over arbitrary rings, so the monomorphism results for T and Ar are characteristic-free; this raises the possibility that the same proof works for other Schur functors rather than only tensor, exterior, and symmetric powers."],"forward_implications":["The classical Segre embedding P(E)×_X P(F)→P(E⊗_{O_X}F) is recovered as the m_1=m_2=1 case of T, and the Plücker embedding Gr(V,m)→P(⋀^m V) as the case r=m of Ar.","For Ar no hypothesis on the base scheme is required, so over any base the Grassmannian of a vector bundle contains the Grassmannian of its r-th exterior power as a closed subscheme.","For Sr the hypothesis m>1 or r-invertible cannot simply be dropped: Remark 3.16 constructs two distinct line subbundles over a ring of characteristic p>0 with p|r whose r-th symmetric powers coincide.","Composing T with the diagonal gives Tr as a closed immersion as well, and the author notes that the same method extends to flag varieties by induction from the Grassmannian case."],"supporting_citations":[{"why":"Supplies the Grassmannian as a fine moduli scheme and the Plücker embedding realizing Grassmannians as closed subschemes of projective space, which yields properness.","marker":"[4]"},{"why":"Supplies the criterion that a proper monomorphism of finite presentation is a closed immersion, the key step that turns injectivity into an embedding.","marker":"[3]"},{"why":"Gives the definition of finite presentation used to verify the hypotheses of the closed-immersion criterion.","marker":"[2]"},{"why":"Provides the functor-of-points description of Grassmannians and the lemma that immersions are monomorphisms, used for the diagonal argument and the monomorphism proofs.","marker":"[11]"},{"why":"Provides background on proper morphisms and the projective-space formulation of the Segre embedding, which the tensor-product map generalizes.","marker":"[1]"}],"fun_headline_variants":["Closed Grassmannian embeddings from tensor operations","Tensor, wedge, symmetric powers yield closed immersions","Grassmannian closed immersions generalizing Segre–Plücker","Morphism closedness via tensor powers on Grassmannians"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The symmetric-power half rests on Lemma 2.7's assertion that two rank-m subbundles with equal r-th symmetric powers must be equal when m≥2 or r is invertible; the proof's coefficient computation contains a 'by symmetry' reduction that is not fully written out, and the lemma's stated ring condition 'Z[rm-2]' appears to be a typo.","fun_headline_variants_meta":{"raw":{"variants":["Closed Grassmannian embeddings from tensor operations","Tensor, wedge, symmetric powers yield closed immersions","Grassmannian closed immersions generalizing Segre–Plücker","Morphism closedness via tensor powers on Grassmannians"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000211,"raw_usage":{"total_tokens":1305,"prompt_tokens":728,"completion_tokens":577,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":344,"completion_tokens_details":{"reasoning_tokens":510}},"tokens_in":344,"tokens_out":577,"duration_ms":6440,"temperature":1.0,"reasoning_tokens":510,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:15:56.151757+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Work out Lemma 2.7 for m=2 and r=2 over a local ring of characteristic 2: write two rank-2 subbundles whose $Sym^{2}$ images in $Sym^{2}$ W agree while the subbundles differ by a nilpotent matrix entry; if such a pair exists, Sr is not a closed immersion, contradicting Theorem 3.15, while if no such pair exists, the coefficient argument is sound.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Grassmannian as a fine moduli scheme and the Plücker embedding realizing Grassmannians as closed subschemes of projective space, which yields properness."},{"cited_title":"´El´ements de G´eom´etrie Alg´ebrique","cited_arxiv_id":null,"evidence_quote":"Supplies the criterion that a proper monomorphism of finite presentation is a closed immersion, the key step that turns injectivity into an embedding."},{"cited_title":"´El´ements de G´eom´etrie Alg´ebrique","cited_arxiv_id":null,"evidence_quote":"Gives the definition of finite presentation used to verify the hypotheses of the closed-immersion criterion."},{"cited_title":"Stacks Project","cited_arxiv_id":null,"evidence_quote":"Provides the functor-of-points description of Grassmannians and the lemma that immersions are monomorphisms, used for the diagonal argument and the monomorphism proofs."},{"cited_title":"´El´ements de G´eom´etrie Alg´ebrique","cited_arxiv_id":null,"evidence_quote":"Provides background on proper morphisms and the projective-space formulation of the Segre embedding, which the tensor-product map generalizes."}],"review_version":1}