REVIEW 2 major objections 5 minor 11 references
On Certain Morphisms between Flag Varieties
T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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…
desk verdict 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. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (2)
- [Lemma 2.7, Props 3.7 and 3.14] 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.
- [Lemma 2.6] 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.
minor comments (5)
- [Introduction and Section 2] 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).
- [Proposition 3.1, proof] 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.
- [Lemma 2.7] 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.
- [Theorem 3.15, proof] 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.
- [Remark 3.16] 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.
Circularity Check
No circularity: the closed-immersion theorem is derived from independent multilinear lemmas, EGA properness and finite-presentation facts, and Stacks Project references, with the author's own prior work cited only as motivation.
full rationale
The paper's central theorem (Theorem 3.15) asserts that the tensor-product, tensor-power, exterior-power, and (under a condition) symmetric-power maps between Grassmannian schemes are closed immersions. The derivation chain is: (1) determinant identities in Proposition 2.1 are proven by a universal-polynomial argument, not by assuming the theorem; (2) Corollaries 2.2-2.4 convert these determinant facts into injectivity/isomorphy statements for tensor, exterior, and symmetric powers; (3) Lemmas 2.5, 2.6, and 2.7 are self-contained module-theoretic comparisons showing that equality of the appropriate power-submodules forces equality of the original submodules; (4) Propositions 3.1, 3.4, and 3.7 sheafify these lemmas over schemes; (5) Propositions 3.13 and 3.14 conclude that the constructed morphisms are monomorphisms; (6) the closed-immersion conclusion follows from the standard EGA/Stacks fact that a proper, finite-presentation monomorphism is a closed immersion, together with the Plucker embedding to see that Grassmannians are proper and of finite presentation. No parameter is fitted to the target, no prediction is generated from the conclusion, and no uniqueness theorem from the author's prior work is imported. The author's works [5], [6], and [7] appear only in the introduction as motivation, not as load-bearing steps in the proof. The skeptic's concern about the hypothesis 'Z[rm-2]-algebra' in Lemma 2.7, Proposition 3.7, and Proposition 3.14 is a correctness or notation issue about what condition is actually needed and proved, not a circularity: the proof does not presuppose the equality of submodules it is trying to establish, nor does it define the objects in terms of the conclusion. Therefore no circular step is present, and the honest score is 0.
Assumptions & free parameters
assumptions (5)
- standard math Diagonalizable matrices are Zariski dense over an algebraically closed field
- standard math A square matrix over a local ring is injective iff its determinant is a non-zero-divisor
- standard math A proper monomorphism of finite presentation is a closed immersion (EGA IV 8.11.5)
- standard math Grassmannian varieties are proper and of finite presentation over the base scheme
- domain assumption The sheafified version of the local lemmas holds for finite locally free sheaves on arbitrary schemes
Cite this review
Pith. "Pith review of On Certain Morphisms between Flag Varieties." pith.science (2026). https://pith.science/paper/U6NCXYZ2
@misc{pith2026190805449,
author = {Pith},
title = {Pith review of: On Certain Morphisms between Flag Varieties},
year = {2026},
howpublished = {\url{https://pith.science/paper/U6NCXYZ2}},
note = {Machine review of arXiv:1908.05449}
}
read the original abstract
The aim of this paper is to construct certain closed embeddings of Grassmannian varieties, using tensor operations on vector bundles. These embeddings generalize Segre and Pl\"ucker morphisms.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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