{"id":"c8e33b19-beb1-4e16-a44e-fc4bc182f384","arxiv_id":"2608.07105","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Successive blow-ups of orthogonal and symplectic local models along Schubert strata yield potentially semi-stable integral models for maximal parahoric PEL Shimura varieties, with unipotence of inertia on nearby cycles.","lead":"This paper constructs regular, potentially semi-stable integral models for a broad class of Shimura varieties with maximal parahoric level at an odd prime, by blowing up local models along Schubert varieties. The construction gives explicit finite extensions of the reflex field and implies that inertia acts unipotently on nearby cycles and l-adic cohomology.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The omitted bilinear-form verification in Lemma 5.9 is the load-bearing point: the inductive blow-up recursion in Prop. 5.4, and hence Theorem 1.2, reduces to the chart isomorphism D_+(y_{i0,j0}) ≅ U(d-1,ℓ+1)^±ε × A^{4d-2}; without it the semi-stability conclusion is unsupported.","rationale":"After reading the paper, the central assertion is Theorem 1.2, and the proof is a concrete chain: functorial reduction (Prop. 1.8/1.9), Hartl's product result (Prop. 1.10), and then the local blow-up computations in Parts 2/3. The local computations are the genuinely new input. The reader's weakest-assumption choice targets Lemma 5.9, and I agree it is the most load-bearing point. The lemma's conclusion is used as the induction step: each blow-up chart must be exactly U(d-1,ℓ+1)^{±ε} × A^{4d-2}, both for the dimension count and for the identification of strict transforms of determinantal loci. The paper explicitly leaves the residual bilinear-form identities unverified ('routine'), so the claim is not machine-checked or independently confirmed. In good faith, I found the chart formulas plausible: the symplectic analogue in Prop. 9.4 gives a detailed elimination, and the orthogonal case likely follows similarly by a signed Schur-complement argument. But 'likely' is not a proof, and this is precisely the kind of step that can hide a sign error or an extra relation. Secondary concerns (dependence on the same-author preprints [41,42,43] for flatness and stratifications, and the brief equivariance argument in Thm. 7.3) exist, but they are conditions on the input rather than internal gaps in the local construction. The concrete test above would settle the main concern: if the chart ideals agree, the concern does not land and the conditional verdict can be upgraded; if they disagree, Theorem 1.2 is currently unproved. Therefore the verdict should remain CONDITIONAL, unchanged from the reader's assessment.","tokens_in":45877,"tokens_out":11387,"duration_ms":103629,"concrete_test":"Verify Lemma 5.9 for d=2 and d=3 with ℓ=0 by direct elimination: in the chart y_{i0,j0}=1, substitute (5.3.3)-(5.3.5) into all residual row/column equations (r_i,r_j)=α^2 δ_{i,j*}, (c_i,c_j)=α^2 δ_{i,j*} and the spin relations, and compare the resulting ideal with that of U(d-1,1)^±ε × A^{4d-2} using a Gröbner basis over F_p or Q_p. Equality of ideals verifies the isomorphism; any residual relation shows the omitted 'automatic' identities fail and the induction in Prop. 5.4 is invalid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 5.3, Lemma 5.9 claims that on the blow-up chart D_+(y_{i0,j0}) (with i0≠i0* and j0≠j0*) the orthogonal relations Y^t H_{2d+1}Y = α^2 H_{2d+1} and Y H_{2d+1}Y^t = α^2 H_{2d+1} force the remaining entries to be determined by (5.3.3)-(5.3.5), and that after setting z_{s,t}=y_{s,t}-y_{i0,t}y_{s,j0} the chart is U(d-1,ℓ+1)^{±ε} × A^{4d-2}. The proof shows the (s,t)-entries of Z satisfy the smaller orthogonal equations, but then states that '(r_{i0*},r_s)=0 for s≠i0, (c_{j0*},c_t)=0 for t≠j0 are automatically satisfied. We omit these routine verifications for brevity.' This is not a cosmetic omission: the strict-transform computation in the proof of Prop. 5.4 identifies the next center V((T0,∧^r Z)) inside D_+(y_{i0,j0}) using precisely this chart isomorphism, and the whole induction on d (and its split orthogonal and symplectic analogues, Prop. 6.2 and 9.2) repeats the same pattern. If the residual identities fail, the chart is a proper closed subscheme of the claimed product, the dimensions no longer match, and the recursion producing A^{2i^2} × Spec O_F[T0,...,Ti]/(π-T0...Ti) at the end of Prop. 5.4 breaks. No computer algebra or independent derivation is supplied for the omitted identities.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for PEL-type Shimura varieties with maximal parahoric level at an odd prime p, explicit (potentially) semi-stable integral models after a finite extension of the reflex field. The main local theorem is Theorem 1.2: under the hypothesis (†), a G_{O_{E'}}-equivariant modification M^ss -> M^loc_{G,μ} ⊗ O_{E'} exists with M^ss semi-stable over O_{E'}. The construction proceeds by blowing up successive Schubert varieties in the special fiber of the canonical local model, with the affine computations reduced to iterated blow-ups of matrix schemes along determinantal ideals. Via the local model diagram, the authors deduce semi-stable integral models S_K^ss of the associated Shimura varieties (Theorem 11.3) and unipotence of inertia on nearby cycles and ℓ-adic cohomology (Theorem 11.7).","tokens_in":46197,"tokens_out":9048,"duration_ms":86515,"significance":"If the main theorem is correct, the paper provides a substantial advance: explicit desingularizations of canonical local models for orthogonal, symplectic, and Weil-restricted PEL data at maximal parahoric level, with the centers described by Schubert varieties and determinantal ideals. The affine-chart reduction and the inductive matrix blow-up method are natural and potentially reusable. The paper is also honest about its scope, handling the ramified-unitary factors only in forthcoming work. The main limitations are that the local-model input (flatness, integrality, and Schubert stratification) is quoted from the same-author preprints [41,42,43], and several key chart identifications are asserted rather than fully verified; these points are discussed in the major comments.","major_comments":[{"comment":"The proof of the chart isomorphism D_+(y_{i0,j0}) ≅ U(d-1,ℓ+1)^{±ε} × A^{4d-2}_{O_F} ends with the sentence: 'The remaining bilinear form identities, i.e., (r_{i0*},r_s)=0 for s≠i_0, (c_{j0*},c_t)=0 for t≠j_0 are automatically satisfied. We omit these routine verifications for brevity.' This is a load-bearing point, not a cosmetic omission. The induction in Proposition 5.4 identifies the next strict transform V(T_0,∧^r Z) precisely by using this chart isomorphism, and the same pattern is used in the split orthogonal case (Lemma 6.4 and Proposition 6.2) and, after iteration, in the symplectic case (Proposition 9.2). If the residual identities are not checked, the closed immersion constructed in Lemma 5.9 need not be surjective, and the dimension argument comparing with the integral target would not force the chart to be the claimed product; the recursion producing A^{2i^2} × Spec O_F[T_0,…,T_i]/(π-T_0⋯T_i) at the end of Proposition 5.4 would then collapse. The missing verification should be supplied in full, either in an appendix or by an independent computer-algebra check, before the main theorem can be considered established.","section":"Section 5.3, Lemma 5.9"},{"comment":"The proof states: 'For each embedding σ=σ^l_j∈Σ, the local model M^loc_i(σ) is isomorphic to the spin local model M^±_i over the ring of integers of L(√-σ(π_1)) by Theorem 4.3.' This is not immediate. M^loc_i(σ) is defined in Definition 3.4 as the schematic closure of the generic fiber of the naive local model M^naive(L[σ],μ[σ]), whose generic fiber is the disjoint union OGr^+ ⊔ OGr^- of the two orthogonal Grassmannian components, while Theorem 4.3 concerns the spin local model M^±_i, a flat closed subscheme representing one component. The text does not justify why the flat closure of the naive local model should be a single spin component rather than the union M^+_i ∪ M^-_i. If it is the union, the semi-stable resolution must be constructed for the union, and Theorem 5.1 or 6.1, which treats a single spin model, does not apply directly. This identification is central to the Weil-restriction case (Theorem 7.3) and therefore to Theorem 1.2 under (†); it needs a rigorous proof or a revised construction.","section":"Section 7, proof of Theorem 7.3"},{"comment":"After passing from the naive splitting-model diagram (3.0.8) to the flat-closure diagram (3.0.9), the text records only that α_1 is an H-torsor and α_2 is H-equivariant; it does not assert that α_2 remains smooth after taking schematic closures. However, in the proof of Theorem 7.3, diagram (7.0.7) labels α_2 and α_2' as smooth, and the 'linear modification' argument uses this smoothness to transfer the modification Z → ∏ M^loc_i to a modification of M^spl_L. Smoothness is not automatic under passing to flat closures, and the text gives no proof that the splitting-model diagram is compatible with the closures in the needed way. Please either prove that α_2 in (3.0.9) is smooth (or étale locally a product), or show explicitly why smoothness is not needed for the linear-modification step. Without this, the Weil-restriction constructions in Theorems 7.3 and 10.2 are not fully justified.","section":"Sections 3 and 7, splitting-model closure"}],"minor_comments":[{"comment":"The definition 'set i_0^* = 2d+1-j' should presumably read i_0^* = 2d+1-i_0; as written, j is not defined there.","section":"Section 6.2, line after Lemma 6.4"},{"comment":"The phrase 'the morphsim γ: M^ss(L,μ)→M^spl_L is G-equivalent' should be 'G-equivariant'.","section":"Section 10, last paragraph of Theorem 10.2 proof"},{"comment":"Definition 2.5(2) appears to define 'normal crossings' twice in consecutive paragraphs: first as simple normal crossings at every closed point, then again via an étale covering. These two notions (normal crossings vs. simple normal crossings) are not the same in the literature, and the text should be reconciled to avoid ambiguity, especially since the main theorems use both properties.","section":"Definition 2.5(2)"},{"comment":"The notation in the displayed isomorphism of Iwahori–Weyl groups, 'fW_G ≃ ... ≃ fW_{G′}', is confusing: the maps denoted f are used for both the apartment isomorphism and the group isomorphism, and the two uses are never defined. The proof would be clearer with distinct names, e.g., f_A and f_W.","section":"Lemma 8.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends essentially on the same-author preprints [41,42,43] for the flatness, integrality, and Schubert stratification of the relevant local models, and on [39, Proposition 5.21] for part (3) of Theorem 11.7. The editor should ensure that these preprints are publicly available and have been independently refereed before acceptance. The technical gaps identified in the major comments—especially the omitted bilinear-form verification in Lemma 5.9 and the unjustified identification of the flat closure of the naive local model with a spin local model in Section 7—are fixable within the manuscript's scope, but they currently prevent verification of the central claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper is a real step forward, not a repackaging. It constructs semi-stable modifications of maximal-parahoric orthogonal and symplectic local models by blowing up Schubert strata, extends to Weil restrictions, and derives the expected inertia unipotence on nearby cycles and l-adic cohomology. The inductive blow-up along determinantal ideals is a genuinely new technique, and the moduli description of symplectic Schubert varieties (Prop 8.9) is concrete and useful on its own. I agree with the reader's verdict: conditional, not suspicious.\n\nThe mathematical core is the affine chart computation in Lemma 5.9. The proof reduces the chart D_+(y_{i0,j0}) to U(d-1, ℓ+1) × A^{4d-2}, but then says the remaining orthogonality identities are \"automatically satisfied\" and omits them. The stress-test is right that this is load-bearing: the inductive recursion in Prop 5.4 and its analogues all lean on this identification. I want to be fair: in the symplectic case (Prop 9.4) the analogous computation is spelled out and works, which makes me lean toward the orthogonal case being genuinely routine. But \"leans toward\" is not a proof, and the paper gives no computer algebra or alternative derivation. A referee should push for these identities to be written out, or at least a precise reference, before the main theorem is fully certified.\n\nTwo smaller soft spots. First, the flatness, integrality, and Schubert stratification are imported from same-author preprints [41,42,43]. The paper is honest about this, and self-citation is not a flaw when the results are real, but it makes the current paper's value conditional on those preprints. Second, in the Weil-restriction applications (Theorems 7.3 and 10.2), the equivariance of the Hartl blow-up sequence under the full group action is argued briefly; I'd like to see that verified in more detail, especially since the group action on the splitting diagram is subtle.\n\nWho is this for? Researchers working on integral models, local models, and Shimura varieties with parahoric level. If the companion preprints hold up and the omitted identities work out, this is a substantial contribution. It deserves a serious referee; desk rejection would be wrong. My recommendation: send to peer review, with a specific instruction to the referee to check Lemma 5.9 (and the analogous Propositions 6.2 and 9.2).","headline":"A serious, explicit construction that likely resolves Conjecture 2.4 for a substantial PEL class, but the referee must check the omitted bilinear-form identities in Lemma 5.9 before the induction is fully certified.","tokens_in":46805,"tokens_out":2666,"would_cite":true,"duration_ms":49982,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G18","14G35","14B05","14M15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that, for PEL-type Shimura varieties with maximal parahoric level at an odd prime, the singularities of the canonical integral model are resolved by explicitly blowing up a chain of Schubert varieties, yielding a…","keywords":["Shimura varieties","semi-stable integral models","canonical local models","parahoric level","Schubert varieties","blow-up resolutions","nearby cycles","PEL type"],"falsifier":"Compute in the affine chart D₊(y_{i₀,j₀}) of U(d,ℓ) for d = 2 (a 5×5 matrix H) the explicit expressions from equations (5.3.3)–(5.3.5) and check whether the relation (r_{i*}, r_s) = 0 for s ≠ i₀ is identically satisfied; if it fails, the chart is not contained in U(d−1,ℓ+1)×$A^{{4d−2}}$ and the claimed semi-stable cover collapses.","tokens_in":30,"feed_emoji":"🔨","tokens_out":9014,"duration_ms":133153,"temperature":0.7,"pith_summary":"The paper establishes that a large class of PEL-type Shimura varieties with maximal parahoric level at an odd prime admit projective semi-stable integral models after a finite extension of the reflex field. The key claim is local: every relevant canonical local model has a group-equivariant modification whose special fiber is a reduced normal-crossings divisor, obtained by explicitly blowing up a chain of Schubert varieties. The construction covers unramified unitary, symplectic, and even orthogonal similitude groups and their Weil restrictions, assuming the algebra factors satisfy condition (†) and there are no ramified unitary factors. If correct, the resulting regular models force the inertia group to act unipotently on nearby cycles and on the ℓ-adic cohomology of the geometric generic fiber.","feed_headline":"Blow-ups yield semi-stable Shimura models at parahoric level","feed_subtitle":"Successive blow-ups along Schubert strata produce regular models; inertia then acts unipotently.","key_machinery":"The argument is carried by a family of auxiliary affine schemes U(d,ℓ), defined by matrix equations XᵗHX = T₀²H, XHXᵗ = T₀²H, π = T₀T₁⋯T_ℓ together with spin-condition relations (with J_{2d} in place of H in the symplectic case), where X is a (2d+1)×(2d+1) or 2d×2d matrix. The load-bearing identity, Lemma 5.9 (and its symplectic analogue Proposition 9.4), says that the affine chart D₊(y_{i₀,j₀}) of the blow-up of U(d,ℓ) along (X,T₀) is isomorphic to U(d−1,ℓ+1) × $A^{{4d−2}}$; the complementary chart D₊(α) is a smooth group scheme times Spec O_F[T₀,…,T_ℓ]/(π − T₀⋯T_ℓ). Iterating this reduction shrinks the matrix size by one at each blow-up and increases ℓ by one, until only a semi-stable product $A^{{2i²}}$ × Spec O_F[T₀,…,T_i]/(π − T₀⋯T_i) remains. This inductive matrix-size reduction is what converts the geometric problem of resolving Schubert singularities into a concrete computation with determinantal ideals.","core_discovery":"On the paper's own terms, the central result is Theorem 1.2: under assumption (†), for a maximal parahoric group scheme G there is a finite extension E'/E and a G_{O_{E'}}-equivariant modification M^ss → M^loc_{G,µ} ⊗_{O_E} O_{E'} such that M^ss has semi-stable reduction over O_{E'}. The modification is a chain of blow-ups along Schubert varieties S_0 ⊂ S_1 ⊂ ⋯ ⊂ S_{d-1} in the special fiber, each center being the strict transform of the previous Schubert variety; in local affine charts, the centers are cut out by the determinantal ideals (π, ∧^{j+1}X). Via the local model diagram, this produces a projective semi-stable model S_K^ss of the Shimura variety S_K over O_{E'}, regular with reduced normal-crossings special fiber, and consequently the inertia group acts unipotently on the nearby cycles and on the ℓ-adic cohomology of the geometric generic fiber. The paper also gives explicit moduli descriptions of the Schubert varieties involved, including a new one in the symplectic case.","pith_inferences":["The chart identity U(d,ℓ) → U(d−1,ℓ+1)×A^{4d−2} suggests a general principle for minuscule local models of classical type: each blow-up of a Schubert stratum decreases the rank parameter while adding normal-crossings coordinates, so semi-stability is forced combinatorially; this principle may extend to the ramified unitary factors excluded by (†).","The explicit finite extension E′ is built only from Galois closures of centers and square roots of uniformizers; one could test computationally whether a smaller extension suffices for concrete examples such as GSp_{2g} with Iwahori level, where earlier constructions of semi-stable models for genus 3 suggest the bound may be sharp.","Because inertia acts unipotently on the cohomology of the geometric generic fiber, the monodromy filtration is the one predicted by the weight-monodromy conjecture; the semi-stable model constructed here gives a place to verify the corresponding weight filtration term by term.","The chain of Schubert varieties used as blow-up centers has the shape of a single chain ordered by inclusion; a testable extension would be to relate the number of blow-ups to the length of this chain and to predict resolutions for other minuscule local models from the same order."],"forward_implications":["After the explicit finite extension E′ described in (11.2.1), the Shimura variety Sh_K(G,X) acquires a regular integral model whose special fiber is a reduced divisor with normal crossings.","For every maximal parahoric level, the canonical local model satisfies the semi-stable resolution conjecture after base change to O_{E′}.","The inertia group of E′ acts trivially on the nearby cycles of the semi-stable model and unipotently on the nearby cycles and ℓ-adic cohomology of the original canonical model and of the geometric generic fiber.","The construction gives an explicit moduli-theoretic description of the Schubert varieties in the special fibers, including a new symplectic description, and shows they form a single chain ordered by inclusion.","The local resolutions are proved uniformly for split and quasi-split orthogonal, symplectic, and Weil-restricted groups, so the same blow-up sequence resolves the corresponding local models after base change to the splitting field."],"supporting_citations":[{"why":"Establishes existence, flatness, projectivity and normality of canonical local models M^loc_{G,µ}, the objects the paper resolves.","marker":"[1]"},{"why":"Supplies flatness, Schubert stratification and explicit affine charts for split even orthogonal spin local models used in Part 2.","marker":"[42]"},{"why":"Supplies the analogous quasi-split orthogonal local model results and the chain of Schubert cells used in the quasi-split blow-up computations.","marker":"[43]"},{"why":"Gives flatness and the Schubert cell stratification of symplectic local models, plus the standard affine chart the symplectic blow-up sequence starts from.","marker":"[13]"},{"why":"Introduces splitting models for Weil-restricted groups, used to reduce the Weil-restriction cases to products of classical local models.","marker":"[30]"},{"why":"Hartl's proposition that a product of semi-stable schemes becomes semi-stable after blowing up products of irreducible components, used for the Weil-restriction arguments.","marker":"[18]"},{"why":"Provides the doubly symplectic standard tableaux basis used to prove the integrality of the local rings R_i(ℓ), hence reducedness of the symplectic Schubert varieties.","marker":"[6]"},{"why":"Constructs the canonical integral model and the local model diagram used to pass from local modifications to global semi-stable models of Shimura varieties.","marker":"[34]"},{"why":"Supplies the theorem that inertia acts trivially on the nearby cycles of a semi-stable scheme, used to deduce the unipotence statements in Theorem 11.7.","marker":"[14]"}],"fun_headline_variants":["Parahoric Shimura models become semi-stable via blow-ups","Blow-ups yield regular semi-stable Shimura models","Inertia unipotent for semi-stable Shimura models","Semi-stable integral models from Schubert blow-ups","Regular semi-stable Shimura models with normal crossings"],"cache_read_input_tokens":48768,"weakest_assumption_plain":"The load-bearing premise is that the omitted verification in Lemma 5.9—that the remaining bilinear-form identities hold automatically in the main affine chart of each blow-up—is correct, since every subsequent chart computation, including the symplectic analogue, depends on it.","fun_headline_variants_meta":{"raw":{"variants":["Parahoric Shimura models become semi-stable via blow-ups","Blow-ups yield regular semi-stable Shimura models","Inertia unipotent for semi-stable Shimura models","Semi-stable integral models from Schubert blow-ups","Regular semi-stable Shimura models with normal crossings"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000927,"raw_usage":{"total_tokens":3960,"prompt_tokens":923,"completion_tokens":3037,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":2954}},"tokens_in":539,"tokens_out":3037,"duration_ms":20371,"temperature":1.0,"reasoning_tokens":2954,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:46:26.898660+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute in the affine chart D₊(y_{i₀,j₀}) of U(d,ℓ) for d = 2 (a 5×5 matrix H) the explicit expressions from equations (5.3.3)–(5.3.5) and check whether the relation (r_{i*}, r_s) = 0 for s ≠ i₀ is identically satisfied; if it fails, the chart is not contained in U(d−1,ℓ+1)×$A^{{4d−2}}$ and the claimed semi-stable cover collapses.","supporting_citations":[{"cited_title":"On $p$-adic integral moduli schemes and local models for PEL type D","cited_arxiv_id":"2602.23813","evidence_quote":"Supplies the analogous quasi-split orthogonal local model results and the chain of Schubert cells used in the quasi-split blow-up computations."},{"cited_title":"Pappas, M","cited_arxiv_id":null,"evidence_quote":"Introduces splitting models for Weil-restricted groups, used to reduce the Weil-restriction cases to products of classical local models."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Hartl's proposition that a product of semi-stable schemes becomes semi-stable after blowing up products of irreducible components, used for the Weil-restriction arguments."},{"cited_title":"de Concini,Symplectic standard tableaux, Adv","cited_arxiv_id":null,"evidence_quote":"Provides the doubly symplectic standard tableaux basis used to prove the integrality of the local rings R_i(ℓ), hence reducedness of the symplectic Schubert varieties."},{"cited_title":"G¨ ortz,Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models forGL 4 andGL 5, J","cited_arxiv_id":null,"evidence_quote":"Supplies the theorem that inertia acts trivially on the nearby cycles of a semi-stable scheme, used to deduce the unipotence statements in Theorem 11.7."}],"review_version":1}