{"id":"85d9fa55-6ba6-439f-8c03-92cd4a367679","arxiv_id":"2411.17487","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper describes cell closures and singular loci of regular Hessenberg varieties for the minimal indecomposable Hessenberg space, generalizing the type A Peterson variety results to all Lie types.","lead":"This paper studies regular Hessenberg varieties in flag varieties, a family that includes the Peterson variety and toric varieties. It computes how the affine cells close up, and it classifies the singular points, including the singular locus of the Peterson variety in every Lie type.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Type C2 is missing from the Figure 1 verification, leaving Theorem 9.2's singular-locus classification unproved for Sp4.","rationale":"The reader's weakest assumption identified the type-by-type verification of Corollary 9.8 as the fragile point in Theorem 9.2. My concern agrees and makes the issue concrete: type C2 is not covered by Figure 1, yet it lies in W* \\ W**, so the proof of Corollary 9.8 is incomplete even if the table is correct. The reader's separate objection to Section 5 is a real bug in the K-theory formulas, but it is not load-bearing for Theorem 9.2 or Theorem 4.6; those geometric results can survive a correction in Section 5. The missing C2 verification, by contrast, directly threatens the paper's main singular-locus theorem. A single computational Jacobian check for Sp4 at ˙s1B would settle whether W* is wrong or merely under-proved. I therefore leave the verdict at CONDITIONAL/UNCHANGED, pending that check.","tokens_in":102,"tokens_out":31192,"duration_ms":446914,"concrete_test":"Compute the Jacobian matrix of the patch ideal I_{s1} for the Peterson variety of Sp4 at ˙s1B using the generators in Lemma 6.2 and the Jacobian criterion of Lemma 6.4. If the rank is 2, the point is smooth and W* for type C2 is wrong; if the rank is less than 2, the point is singular and the theorem's statement survives but its proof needs a separate argument for this case. Independently, enumerate all K in W* \\ W** for every irreducible type with a root-system package and verify the two conditions of Proposition 9.7; this will confirm that no admissible pair exists for type C2, isolating the missing case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 9.2 asserts the singular locus of P∆ is exactly the union over w ∈ W* of the Peterson-Schubert cells. The proof of Corollary 9.8 splits into W** cases handled by Proposition 9.3 and W* \\ W** cases handled by Proposition 9.7, with Figure 1 said to verify the latter and the actual check outsourced to diagrams in [EHP14]. This is the load-bearing step: if Figure 1 misses a case, the claimed set W* is either incomplete or incorrect. There is a concrete omitted case. For type C2 (G = Sp4), W* = {∅, {α1}, {α2}} while W** excludes K = ∆ \\ {α2} = {α1}; thus K = {α1} lies in W* \\ W**. But Figure 1 only lists type C_n for n ≥ 3. If one tries to apply the table's formula at n = 2, the proposed roots specialize to η2 = -α1, which lies in Φ^-_K, violating the hypotheses of Proposition 9.7. Consequently the paper gives no argument that the cell C_{s1} ∩ P∆ is singular, and if ˙s1B is in fact smooth, then W* is wrong for type C2. Since the type B2 classification excludes the corresponding Hermitian-type K, the discrepancy is not merely a presentational gap: it may signal a misclassification. This is precisely the diagram-check risk flagged by the reader, sharpened to a specific rank-two case absent from the table.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies regular Hessenberg varieties attached to the minimal indecomposable Hessenberg space H∆ in a complex reductive flag variety. The main structural result (Theorem 4.6) identifies the closure of each nonempty Hessenberg–Schubert cell with a regular Hessenberg variety in the flag variety of a Levi subgroup, leading to inclusion relations, K-theory and cohomology formulas, and smoothness criteria. The later sections analyze singularities: Theorem 7.1 reduces singularity of a Weyl flag in Hess(XJ) to the same question on a Peterson variety of a Levi subgroup, Theorem 8.5 gives the type-A classification of smooth permutation flags, and Theorem 9.2 claims a complete description of the singular locus of the Peterson variety in every Lie type as a union of Peterson–Schubert cells.","tokens_in":41627,"tokens_out":17608,"duration_ms":167853,"significance":"The paper attacks a natural and important problem: the global singular locus of Peterson and other regular Hessenberg varieties. The cell-closure theorem (Theorem 4.6) and the reduction theorem (Theorem 7.1) are valuable and are proved by detailed, mostly standard patch-ideal and Jacobian arguments. The type-A classification of smooth permutation flags and the Hessenberg–Schubert smoothness criterion (Theorem 9.11) are appealing and useful if the supporting results hold. The manuscript contains several machine-checkable or readily checkable combinatorial computations, and the authors are careful to credit prior results on reducedness of patch ideals. However, two load-bearing issues—an omitted and apparently false type-C2 case in Theorem 9.2, and a rational-coefficient error in the K-theory formula of Section 5—require substantial correction before the paper's central claims can be accepted.","major_comments":[{"comment":"The type C2 case is omitted from Figure 1, and Theorem 9.2 appears to be false for G=Sp4 as stated. In Definition 9.1, for type Cn the set W* contains all proper K, while W** excludes K=∆∖{αn}; for n=2, K={α1} lies in W*∖W**. Figure 1 only lists Cn for n≥3, so Corollary 9.8 gives no argument for this cell. A direct application of the paper's own Lemma 6.6 at w=s1 shows that ˙s1B is smooth: w(Φ−∖Δ−)={−α2, −(α1+α2)}, and the two patch generators have independent linear terms in z_{−(α1+α2)} and z_{−(2α1+α2)}, so the Jacobian has rank equal to codim(P∆)=2. Thus C_{s1}∩P∆ is not contained in the singular locus, contradicting Theorem 9.2 for type C2. The type C2 case must be added to the table and W* must be adjusted (or the present computation refuted), before the singular-locus theorem can be regarded as proved in all Lie types.","section":"Section 9, Definition 9.1 and Figure 1"},{"comment":"The K-theory formulas contain a rational coefficient that is not an integral K-class. For G=GL2(C), J=∅, and w=e, the Hessenberg–Schubert variety is the reduced point ˙eB, whose class in K0(P1) is 1−[L_{−α}]. Corollary 5.4 gives (1/2)(1−[L_{−α}]); this is not a K-class of a subvariety. The same factor |WDes(w)|/|W| appears in Corollary 5.6. Lemma 5.3 therefore does not compute [OBL] in K0(B) as stated; either the factor is incorrect or the statement needs to specify a different ring (for example, rationalized K-theory). Since Corollaries 5.4 and 5.6 are stated as plain K0(B)/H*(B) identities, this is a load-bearing error in Section 5.","section":"Section 5, Lemma 5.3 and Corollary 5.4"}],"minor_comments":[{"comment":"The sentence contains the duplicated word 'although although'.","section":"Example 7.5"},{"comment":"The phrase 'Hesssenerg-Schubert varieties' contains a typo and should read 'Hessenberg–Schubert varieties'.","section":"Remark 4.7"},{"comment":"The auxiliary root γ is used in the table but is not defined in the caption or the surrounding text; each row should specify how γ is chosen.","section":"Figure 1"},{"comment":"If the K-theory formulas are intended in a localized or rationalized version of K0(B), this should be stated explicitly before the corollaries are used.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The type-C2 issue is the most serious. It is not merely a missing diagram check: a direct Jacobian computation with the paper's own Lemma 6.6 indicates that ˙s1B is smooth for Sp4, so the W* of Definition 9.1 appears to be wrong for C2. If the authors confirm this and add the appropriate exception, the main theorem may survive in modified form. The K-theory coefficient problem is also genuine and should be corrected before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper has two genuine problems: the Section 5 formulas are off by rational coefficients, and the proof of the all-types Peterson singular locus (Theorem 9.2) misses type C2. The structural geometry is mostly right, and with fixes this deserves a serious referee.\n\nThe genuinely new and valuable part is Theorem 4.6: the closure of each Hessenberg–Schubert cell in a regular Hessenberg variety is isomorphic to a regular Hessenberg variety in a Levi flag variety. That is a clean, useful structural result, and it drives the type A smoothness classification (Theorem 8.5) and the root-theoretic smoothness criterion (Theorem 9.11), both of which look correct. The patch-ideal/Jacobian arguments are standard and carefully done, and the paper is honest about what is open.\n\nNow the soft spots. First, Section 5. For G=GL2, J=∅, w=e, Corollary 5.4 gives the class (1/2)(1-[L_{-α}]) for a point in P^1. That is not an integral K-theory class. Either the |W_Des(w)|/|W| normalization is wrong or the intended statement is in K^0⊗Q. Corollary 5.6 inherits the problem, so the Peterson basis classes are also suspect. This is a real error, not a typo.\n\nSecond, and more serious, Theorem 9.2's proof has a genuine gap. Corollary 9.8 says that every w in W* \\ W** is handled by Figure 1, but Figure 1 has no row for type C2. For Sp4, take K={α1}; this lies in W* \\ W**, so the theorem says the corresponding cell is singular. The C_n formulas in Figure 1 do not specialize to n=2: the required η2 would lie in Φ^-_K, violating the hypotheses of Proposition 9.7. So the paper gives no argument for that cell. Since C2 is isomorphic to B2, and the B2 classification treats the corresponding K as smooth, this is not just a missing line in a table—the classification could be wrong. The authors need to add C2 or handle it directly. I also think the verification of Figure 1 is too thin: outsourced to diagrams in [EHP14] with no case-by-case check in the paper.\n\nBottom line: this is serious, well-written work, and the core ideas are likely right. But as it stands, the K-theory formulas are wrong and the Peterson singular-locus classification is unproved for a rank-two case. Both are fixable, but they need to be addressed. I would send it to a referee and expect a revision, not an acceptance.","headline":"Strong structural results, but the K-theory formulas are off by rational coefficients and the proof of the all-types Peterson singular locus misses type C2; both need fixing before acceptance.","tokens_in":42181,"tokens_out":9027,"would_cite":false,"duration_ms":76891,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14M15","14B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the singular locus of the Peterson variety is a union of smaller Peterson varieties in all Lie types, and that every Hessenberg–Schubert cell closure is a regular Hessenberg variety in a Levi flag variety.","keywords":["regular Hessenberg varieties","Hessenberg–Schubert varieties","Peterson variety","singular locus","flag varieties","Weyl group","toric varieties","K-theory"],"falsifier":"In type D_n with n ≥ 4, set K = ∆ \\ {α1} and compute the patch ideal of the Peterson variety at the Weyl flag y_K B. Theorem 9.2 predicts that every point of C_{y_K} ∩ P_∆ is singular, so the Jacobian matrix of the two generators singled out by Proposition 9.7 must drop rank at the origin; one full-rank matrix there would refute the theorem.","tokens_in":41115,"feed_emoji":"📐","tokens_out":9058,"duration_ms":75578,"temperature":0.7,"pith_summary":"This paper studies the regular Hessenberg varieties attached to the minimal indecomposable Hessenberg space, the smallest subspace choice that still keeps the geometry connected. It proves that the closure of every Hessenberg–Schubert cell is isomorphic to a regular Hessenberg variety in the flag variety of a Levi subgroup, giving a complete description of how the cells fit together. It then pins down singularities: the singular Weyl flags in any such variety are controlled by the Peterson variety of a Levi, and the full singular locus of the Peterson variety is a union of smaller Peterson varieties in all Lie types. A reader should care because the Peterson variety sits inside the quantum cohomology story of flag varieties, and knowing exactly where it is singular is the missing piece for many Schubert-type calculations.","feed_headline":"Singular Peterson loci are smaller Peterson varieties in all Lie types","feed_subtitle":"Cell-by-cell analysis reduces regular Hessenberg varieties to Levi flag varieties and locates every singular point.","key_machinery":"The central machinery is the minimal indecomposable Hessenberg space H_∆ = b ⊕ ⊕_{α∈∆} g_{−α} together with the regular elements X_J = S_J + N_J indexed by subsets J of simple roots. The argument runs through the reduced decomposition w = τ_w y_Des(w), which sends the closure of a Hessenberg–Schubert cell into the flag variety of the Levi subgroup L_w, and through shifted patch ideals I_{w,J} whose generators are root-space projections; the Jacobian criterion on these local defining equations converts singularity into rank drop. The final classification rests on the root-theoretic sets W^* and $W^{{**}}$, defined by deleting certain simple-root subsets from the Weyl group, and on the type-by-type table of root posets used to verify Proposition 9.7.","core_discovery":"On the paper's own terms, the discovery is that the affine-cell structure and singular geometry of Hess(X_J) reduce to smaller flag varieties. Theorem 4.6 identifies C_w ∩ Hess(X_J) with Hess_{L_w}(X_{J,w}) inside the flag variety of the Levi subgroup L_w determined by the descent set of w; consequently every Hessenberg–Schubert variety is itself a regular Hessenberg variety. Theorem 7.1 reduces singularity of the Weyl flag ẇB to a singularity question in the Peterson variety of a Levi subgroup, and Theorem 9.2 completes the Peterson picture: for a simple algebraic group, the singular locus of P_∆ is the union over w ∈ W^* of the Peterson–Schubert cells C_w ∩ P_∆, with W^* given by simple root data. In particular the singular locus of every Peterson variety is a union of smaller-dimensional Peterson varieties, and all regular Hess(X_J) outside the toric case are singular.","pith_inferences":["This paper's Levi-reduction theorem suggests that the full singular locus of Hess(X_J), which remains open outside the nilpotent and semisimple cases, should be describable as a union of translated Peterson-type loci; Example 7.4 already shows it need not be a union of Hessenberg–Schubert cells, so the natural test is to compare Jacobian ranks at non-Weyl points in type A.","The pattern avoidance in Theorem 8.5 and the bracket condition [v^{-1}(K), Des(w)] = ∅ in Theorem 9.11 are two languages for the same smoothness statement; a direct translation for types B, C, and D would yield explicit pattern-avoidance lists, and one could check them against the existing conjectural lists for GL_n in the literature.","Because the K-class formula in Corollary 5.4 is independent of J, the equivariant cohomology of any regular Hess(X_J) should admit a basis indexed by J-admissible elements with structure constants governed only by descent sets; this points toward a uniform Schubert-calculus formalism for the whole flat family."],"forward_implications":["Every Hessenberg–Schubert variety in Hess(X_J) is itself a regular Hessenberg variety in a smaller flag variety, so cell-closure inclusion relations are read off from descents and cosets w W_Des(w).","The K-class and cohomology class of each Hessenberg–Schubert variety have closed formulas depending only on Des(w), independent of J, and recover the known Peterson classes.","The singular locus of the Peterson variety is a union of Peterson–Schubert cells, and is controlled entirely by the proper subsets W^* of simple roots; in particular it is a union of smaller Peterson varieties.","For every reductive group whose root system is irreducible of rank at least 2, Hess(X_J) is smooth only in the toric case J = ∅.","In type A, the singular permutation flags are classified by µ-block permutations plus pattern avoidance of 123 and 2143, with an explicit count of smooth permutation flags."],"supporting_citations":[{"why":"Classifies singular points of the type A Peterson variety, the base case that Theorem 8.5 extends and Theorem 9.2 generalizes.","marker":"[IY12]"},{"why":"Supplies the reducedness of patch ideals and the exact sequence that underpin the Jacobian-criterion and K-theory arguments.","marker":"[AFZ20]"},{"why":"Provides affine pavings of Hessenberg varieties in all Lie types, giving the Hessenberg–Schubert cells used throughout.","marker":"[Pre13]"},{"why":"Introduced the affine paving of Hessenberg varieties in type A that motivates the cell decomposition.","marker":"[Tym06]"},{"why":"Defines Hessenberg varieties and proves the regular semisimple case is smooth, the toric baseline for Corollary 9.10.","marker":"[DMPS92]"},{"why":"Its appendix diagrams are the basis for the type-by-type verification in Corollary 9.8 that Figure 1 covers all non-chain root posets.","marker":"[EHP14]"},{"why":"Identifies the dense Peterson–Schubert cell as a Z_G(N)-orbit, proving the longest Weyl group element is a smooth point.","marker":"[Bal17]"},{"why":"Provides the root-system conventions and highest-root table used in Lemma 9.5 and the W^* definitions.","marker":"[Hum78]"}],"fun_headline_variants":["Peterson singular loci are unions of smaller Peterson varieties","Only toric regular Hessenberg varieties are smooth","Hessenberg–Schubert varieties are regular Hessenberg in Levi flags","Singularity of all regular Hessenberg varieties reduces to Peterson cells"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the type-by-type table in Figure 1, verified from the diagrams in [EHP14], really lists every subset K for which Φ^+ \\ Φ^+_K is not a chain; if a case is missing, the announced set W^* omits a singular stratum.","fun_headline_variants_meta":{"raw":{"variants":["Peterson singular loci are unions of smaller Peterson varieties","Only toric regular Hessenberg varieties are smooth","Hessenberg–Schubert varieties are regular Hessenberg in Levi flags","Singularity of all regular Hessenberg varieties reduces to Peterson cells"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000972,"raw_usage":{"total_tokens":4141,"prompt_tokens":963,"completion_tokens":3178,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":3105}},"tokens_in":579,"tokens_out":3178,"duration_ms":22649,"temperature":1.0,"reasoning_tokens":3105,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:06:13.487180+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In type D_n with n ≥ 4, set K = ∆ \\ {α1} and compute the patch ideal of the Peterson variety at the Weyl flag y_K B. Theorem 9.2 predicts that every point of C_{y_K} ∩ P_∆ is singular, so the Jacobian matrix of the two generators singled out by Proposition 9.7 must drop rank at the origin; one full-rank matrix there would refute the theorem.","supporting_citations":[],"review_version":1}