{"id":"a0c87cc4-8844-452c-a5f5-1a28526c4b0b","arxiv_id":"2504.18293","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any finite abelian groups B0 ⊂ B, there exists a smooth rationally connected variety over a number field whose unramified Brauer group is B and whose Hasse principle obstruction is precisely B0.","lead":"The paper builds algebraic varieties whose hidden obstruction group, the Brauer group, can equal any prescribed finite abelian group, with exactly a prescribed subgroup causing the failure of the Hasse principle. It settles two open questions in arithmetic geometry about the limits of the Brauer-Manin obstruction.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The unproved assertion in Remark 5.7 that the pulled-back model X is very good carries the Brauer-group identification; without a proof of Definition 3.2(4), Theorem 5.4(1) is not supported.","rationale":"After reading the paper in good faith, I find the main construction coherent and the claimed theorem plausible. The most load-bearing weakness is exactly the one the Reader identified: the unproved assertion in Remark 5.7 that the pulled-back model X is a very good model. The final isomorphism B ≃ Br(X) in Theorem 5.4, and hence the precise obstruction in Theorem 5.1, depend on Theorem 4.1, whose hypotheses require Definition 3.2(4). The paper's own remark concedes the gap. I considered whether other issues—for instance the effect of deg(h) on the residue at infinity, or the existence of archimedean local points after pull-back—are more serious; both appear addressable (multiples of n vanish in H^1(k,Q/Z), so the Brauer-group formula is stable, and h can presumably be chosen with appropriate real behaviour), and the text at least gestures at the latter. The VAV15 assertion is probably true and likely fillable by a direct gluing-chart comparison, but it is not a one-line consequence, so the correct verdict is conditional acceptance pending that proof. This does not change the Reader's verdict.","tokens_in":21808,"tokens_out":22913,"duration_ms":255164,"concrete_test":"Prove or disprove Definition 3.2(4) for X = X' ×_{P^1,h} P^1: compute Pic of the geometric generic fibre of the VAV15 model for P'∘h either by explicitly writing its gluing charts for N_{K/k}(z)=P'(h(x)) and comparing with the pull-back, or by proving that base change along the finite extension k(x')→k(x) sends Pic(X'_{\\bar{k}(x')})=Z to Pic(X_{\\bar{k}(x)})=Z with trivial Γ-action. If the verification fails for some h produced by Proposition 2.10, Theorem 5.4(1) collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.4 constructs X = X' ×_{P^1,h} P^1 and uses it to conclude B ≃ Br(X) and to control local images. The identification B ≃ Br(X) rests on Theorem 4.1, which is stated only for very good models, i.e. satisfying Definition 3.2(4) (Pic(X_{\\bar{k}(x)}) = Z with trivial Galois action). The paper explicitly declines to verify this for X in Remark 5.7: \"We do not prove that the pull-back X of X′ by h is a very good model... But we are not going to present the details.\" Conditions (1)-(3) are at least plausibly supplied by h being étale over the degeneracy locus and by h(∞)=∞, but (4) requires a separate argument: the base change of the generic fibre by the extension k(x')→k(x) along h must preserve the geometric Picard group and the triviality of the Galois action. This is true in the present setting, but it is not automatic, and it is load-bearing: if X were merely a good model, Br_vert(X) could be a proper subgroup of Br(X), so the isomorphism B ≃ Br(X) in conclusion (1) would not follow, and the precise obstruction statement in Theorem 5.1 would lack its stated basis.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, over a number field k containing a primitive n-th root of unity, smooth rationally connected varieties X with prescribed finite unramified Brauer group B (modulo constant classes) and with a prescribed nonzero subgroup B0 that precisely obstructs the Hasse principle: for every subgroup B′ of B, the Brauer–Manin set X(A_k)^{B′} is empty if and only if B0⊂B′. The varieties are smooth compactifications of cyclic normic bundles N_{K/k}(z)=P(x) over P^1, with K/k cyclic of degree n. The proof has two main parts: first, a computation of the unramified Brauer group of such normic bundles (Theorems 3.6 and 4.1), carried out with the smooth models of Várilly-Alvarado and Viray and expressed in terms of the factorization of P; second, an adaptation of the method of Berg–Pagano–Poonen–Stoll–Triantafillou–Viray–Vogt, pulling back a base model along an endomorphism h of P^1 to control the local evaluations of the Brauer classes (Theorem 5.4). The main theorem (Theorem 5.1) follows from these ingredients, together with a lemma showing that the image of the local evaluation maps can be prescribed exactly.","tokens_in":22061,"tokens_out":32663,"duration_ms":304902,"significance":"If the identified gap is repaired, this is a substantial contribution. The main result simultaneously answers Questions 1 and 2 from the introduction, providing rationally connected varieties with arbitrary finite unramified Brauer groups and with arbitrary prescribed minimal obstructing subgroups. The Brauer-group computation is detailed, self-contained (conditional on the VAV15 model), and generalizes earlier results of Várilly-Alvarado–Viray and Skorobogatov. The obstruction part extends the recent work [BPP+24] from conic bundles to higher-dimensional normic bundles and from elementary 2-groups to arbitrary finite abelian groups. The paper also provides explicit examples and points out an inaccuracy in [VAV12, Theorem 3.2]. The main weakness is the unproved assertion about the pulled-back model in Remark 5.7, which is load-bearing for the central theorem.","major_comments":[{"comment":"The proof of Theorem 5.4 uses the pullback X = X′ ×_{P^1,h} P^1 and asserts, without proof, that X is a very good model of the normic equation N_{K/k}(z)=P′(h(x)). This is load-bearing: Theorem 4.1, which provides the isomorphism B≃Br(X) in conclusion (1), is stated only for very good models, and Definition 3.2(4) (Pic(X_η)=Z with trivial Galois action) is not automatically preserved under the base change of the generic fiber by the finite extension k(x′)→k(x) attached to h. The authors explicitly write “We do not prove that the pull-back X of X′ by h is a very good model... But we are not going to present the details.” Since the central theorem depends on this point, the missing verification must be supplied (or a precise reference covering exactly this pullback situation must be given).","section":"§5.4, Remark 5.7"},{"comment":"The sentence “Theorem 4.4 together with Remark 4.3 allows us to conclude that the composition B→Brnr(X′)→Brnr(X) is a monomorphism ... and induces an isomorphism onto Brnr(X)” invokes Theorem 4.1 for the model X. Without the missing proof that X is very good, only the vertical Brauer group Br_vert(X) is controlled by the preceding arguments; the isomorphism B≃Br(X) in conclusion (1) would not follow. This is the same gap as in Remark 5.7, but it is worth flagging the specific sentence where the load-bearing invocation occurs.","section":"§5.4, paragraph after defining X"}],"minor_comments":[{"comment":"The statement says “for t=1,...,r, the image Im(λ_{w_r}:X(k_{w_r})→\\hat B)=Λ_r”; the subscript should be w_t on both the place and the set, not w_r.","section":"§5.2, Theorem 5.4(2)"},{"comment":"The symbol c appears in the statements of Theorems 3.6(4) and 4.1 (“if c is further assumed to be a norm for K/k”) but is not defined in those statements; it is the leading coefficient of P from equation (3.1). Please add the definition.","section":"§3.3 and §4.1, Theorems 3.6(4) and 4.1"},{"comment":"The phrase “Let m be positive integers” should be “Let m be a positive integer.”","section":"§5.3, Lemma 5.5"},{"comment":"The sentence “A further change of variable z↦z(x+c)^{m+1} does not make any affect on the computation of unramified Brauer groups since all the generators are vertical, i.e. independent of z” is vague; a short justification that the change of variables induces an isomorphism of the relevant models would improve clarity.","section":"§5.4, proof after the automorphism step"},{"comment":"The proof of surjectivity in Proposition 5.6(3) requires the reductions of Q′_i modulo w_t to be separable and pairwise coprime. The advanced version of Lemma 4.5 supplies roots outside a given finite set, but the text does not explicitly state that the finite sets can be chosen to include the roots of the other Q′_j and that this is sufficient; the argument is likely correct but should be spelled out.","section":"§5.3, proof of Proposition 5.6"}],"recommendation":"major_revision","confidential_remarks":"The main gap is localized to the unproved assertion in Remark 5.7. From the structure of the proof, the assertion appears true: the generic fiber of the pullback X is isomorphic to the generic fiber of the VAV15 model for P′(h(x)), and the other conditions of Definition 3.2 can be checked from h being étale over the degeneracy locus and from the behavior at infinity. Thus the paper is likely repairable. My recommendation of major revision is based on the fact that the missing proof is load-bearing for the central theorem, not on any suspicion that the statement is false. I would ask the authors to either prove the assertion or cite an existing result that covers exactly this pullback situation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one. Liang and Liu prove that for any finite abelian group B and any nonzero subgroup B0, over a number field k containing a primitive n-th root of unity, there exists a smooth rationally connected k-variety X with Br(X)/Br(k) ≅ B and with B0 the smallest subgroup of B whose Brauer–Manin set is empty. That answers two natural open questions. Previous results only handled p-groups, small groups, or elementary abelian 2-groups. The construction uses normic bundles over P^1; nothing about it is ad hoc.\n\nThe Brauer group computation in Sections 3–4 is the strongest part. Theorem 4.1 gives an explicit formula for the unramified Brauer group of a cyclic normic equation in terms of factorization data, generalizing VAV12 and removing their technical assumptions. The example in Remark 4.2 is a useful caution. Theorem 4.4, realizing every finite abelian B as the Brauer group, is a clean application of the formula plus the group-theoretic lemmas.\n\nThe obstruction part in Section 5 follows the BPP+24 template: control local evaluation maps, use a finite field counting lemma, then pull back by a polynomial h to concentrate the images. The finite field lemma (Lemma 5.5) uses a Hasse–Weil bound; the genus computation looks fine to me.\n\nNow the soft spot, which the reader's report flagged. Remark 5.7 says the authors do not prove the pulled-back model X = X' ×_{P^1,h} P^1 satisfies the 'very good' condition (4), i.e. the geometric Picard group of the generic fiber is Z with trivial Galois action. The stress test suggested this is load-bearing because Theorem 4.1 is stated for very good models. I think that concern is misplaced. X is a smooth projective model of the new normic equation N(z)=P'(h(x)), and unramified Brauer groups are birational invariants for smooth projective varieties. Apply Theorem 4.1 directly to the polynomial P'∘h, using VAV15's existence of a very good model; the factorization pattern of P'∘h is verified in the proof, so the same computation gives Br(X) ≅ B. You don't need to prove condition (4) for this particular X. The authors should just say that, replacing the hand-wavy claim in Remark 5.7. As written, the remark is a genuine exposition gap but not a mathematical one. The rest of Section 5 doesn't need condition (4) either; the classes involved are vertical by construction.\n\nSecondary nit: Remark 4.2's criticism of VAV12 is brief; a referee might want more detail, but the counterexample is concrete. The paper leans heavily on VAV15 and BPP+24, but that's normal for this area.\n\nBottom line: this deserves serious refereeing. With a small revision to Remark 5.7 and a few clarifying sentences, it should be accepted. I'd bring it to reading group and would cite it in my own work.","headline":"The paper delivers: arbitrary finite abelian Brauer groups and precise minimal Hasse-principle obstructions, on rationally connected varieties, and the one gap the reader flagged is an exposition gap, not a load-bearing flaw.","tokens_in":22620,"tokens_out":12752,"would_cite":true,"duration_ms":133340,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14G12","14G05","14F22"],"pacs":[],"model":"deepseek-v4-flash","headline":"For any finite abelian group B and any subgroup B0, a rationally connected variety exists whose unramified Brauer group is B and whose Hasse-principle obstruction is exactly B0.","keywords":["Brauer–Manin obstruction","Hasse principle","unramified Brauer group","normic bundles","rationally connected varieties","finite abelian groups","cyclic extensions"],"falsifier":"Specialize to $k=\\mathbb{Q}$, $n=2$, $B=\\mathbb{Z}/2$, and $B_0=B$, choose $a$ and $P'$ as in Proposition 5.6, and choose $h$ as in Proposition 2.10. Compute directly the unramified Brauer group and the local images of the pulled-back model $X=X'\\times_{\\mathbb{P}^1,h}\\mathbb{P}^1$; if either differs from $B$ or from the prescribed set $\\Lambda$, the model assertion on which Theorem 5.1 rests is false.","tokens_in":21529,"feed_emoji":"🎯","tokens_out":14933,"duration_ms":129621,"temperature":0.7,"pith_summary":"This paper proves that the Brauer–Manin obstruction can be given any prescribed finite abelian shape. For any finite abelian group $B$ of exponent dividing $n$, any nonzero subgroup $B_0\\subset B$, and any number field $k$ containing a primitive $n$-th root of unity, the authors construct a smooth rationally connected variety $X$ over $k$ whose unramified Brauer group, modulo constant classes, is isomorphic to $B$. The variety violates the Hasse principle in a precise way: for every subgroup $B'\\subset B$, the Brauer–Manin set $X(\\mathbb{A}_k)^{B'}$ is empty precisely when $B_0\\subset B'$. Thus $B_0$ is the smallest subgroup that accounts for the failure, and the examples are compactified normic bundles defined by $N_{K/k}(z)=P(x)$ with $K/k$ cyclic.","feed_headline":"Every finite abelian group can be exactly the Brauer obstruction","feed_subtitle":"Compactified normic bundles realize any prescribed finite Brauer group and make a chosen subgroup the minimal obstruction to rational…","key_machinery":"The load-bearing objects are normic bundles: affine hypersurfaces $N_{K/k}(z)=P(x)$, where $N_{K/k}$ is the norm of a cyclic extension $K/k$, together with their very good smooth compactifications (proper models satisfying four conditions, including a non-degenerate fibre at infinity and geometric Picard group $\\mathbb{Z}$ on the generic fibre). Three mechanisms carry the argument. First, the vertical Brauer group is computed explicitly as the group of classes $\\chi\\cup(P_i)_n$ modulo a cyclic diagonal kernel, encoded in Theorem 4.1, which turns factorization data of $P$ over $K$ into the prescribed group $B$. Second, a finite-field counting lemma and the local-invariant formula for cyclic algebras allow the construction to force chosen values of the Brauer–Manin pairing at given places. Third, a polynomial endomorphism $h$ of $\\mathbb{P}^1$, produced by a weak-approximation and irreducibility lemma, pulls the model back so that the forced local values become actual local images, while the factorization pattern of $P'(h(x))$ keeps the Brauer group isomorphic to $B$.","core_discovery":"The central discovery is a method for building such varieties rather than an isolated example. The paper expresses $B$ as a quotient of a direct sum of cyclic groups, chooses a cyclic degree-$n$ extension $K=k(\\sqrt[n]{a})$, and selects monic polynomials $P_i=(x^{n_i}-u_i)^{n/n_i}-a$ that are irreducible over $k$ and split over $K$ into equal-degree factors. For $P=\\prod_i P_i$, the unramified Brauer group of a smooth compactification of $N_{K/k}(z)=P(x)$ is exactly $B$, and the relevant classes are vertical, pulled back from the base $\\mathbb{P}^1$. In the second half, the paper generalizes the technique of prescribing local images of Brauer classes: a polynomial endomorphism $h$ of $\\mathbb{P}^1$ is chosen so that, after pulling the whole model back along $h$, the local evaluations of the generating classes take exactly a prescribed set at chosen places and vanish elsewhere. Choosing that set to be the complement of the kernel attached to $B_0$ yields the precise obstruction statement of Theorem 5.1.","pith_inferences":["Lemma 4.5 supplies infinitely many parameter choices, so the same construction should give many non-isomorphic rationally connected varieties realizing a fixed pair $(B_0,B)$; a moduli or field-counting study could make this precise.","Because the local images are controlled through one endomorphism of $\\mathbb{P}^1$, a natural extension is to prescribe arbitrary finite subsets of the dual group at several places simultaneously, producing varieties with a prescribed adelic obstruction pattern rather than a single subgroup.","The unproved assertion in Remark 5.7 is the only visible gap: if the pull-back property can be proved directly or replaced by a purity argument, the main theorem would not depend on an unverified model claim."],"forward_implications":["Every finite abelian group of exponent dividing $n$ is realized, over any number field containing a primitive $n$-th root of unity, as the quotient unramified Brauer group of a rationally connected variety.","The Brauer–Manin obstruction can require an arbitrary prescribed finite abelian subgroup as the minimal obstructing subgroup, going beyond the elementary abelian $2$-groups previously known.","When $B_0=B$, a rationally connected variety exists whose whole Brauer group is necessary and sufficient for the obstruction: no proper subgroup annihilates the Brauer–Manin set.","The constructed varieties have points in every completion, so the failure of the Hasse principle is purely a Brauer–Manin obstruction rather than a local failure."],"supporting_citations":[{"why":"Provides the very good smooth compactification of the normic equation on which the Brauer-group computation is built.","marker":"[VAV15]"},{"why":"Supplies the technique of prescribing local Brauer images through polynomial endomorphisms, including the irreducibility and finite-field lemmas reused here.","marker":"[BPP+24]"},{"why":"Supplies the cohomology of Brauer groups, residue maps, and the exact sequences used to identify vertical Brauer groups.","marker":"[CTS21]"},{"why":"Records the known $n=2$ computation that the general formula of Theorem 4.1 must match.","marker":"[Sko01]"},{"why":"Gives the earlier degree-$p$ computation of these Brauer groups, which the paper generalizes.","marker":"[VAV12]"},{"why":"Provides the local-invariant formula for cyclic algebras used to force prescribed evaluations of Brauer classes at chosen places.","marker":"[Ser79]"},{"why":"Introduces the Brauer–Manin pairing whose vanishing defines the obstruction being prescribed.","marker":"[Man71]"}],"fun_headline_variants":["Prescribe both Brauer group and minimal Hasse obstruction","For any nested finite abelian groups, construct exact Brauer and Hasse obstructions","Prescribe unramified Brauer group and minimal Hasse obstruction","Exact Brauer groups and precise Hasse obstructions on normic bundles","Normic bundles: prescribe Brauer groups and minimal Hasse obstructions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on an assertion it does not prove: after the chosen endomorphism $h:\\mathbb{P}^1\\to\\mathbb{P}^1$ pulls the known smooth model $X'$ back, the new variety $X=X'\\times_{\\mathbb{P}^1,h}\\mathbb{P}^1$ is again the very good model of the normic equation $N_{K/k}(z)=P'(h(x))$; the isomorphism $B\\simeq \\overline{\\mathrm{Br}}(X)$ and the control of local images both use that model property.","fun_headline_variants_meta":{"raw":{"variants":["Prescribe both Brauer group and minimal Hasse obstruction","For any nested finite abelian groups, construct exact Brauer and Hasse obstructions","Prescribe unramified Brauer group and minimal Hasse obstruction","Exact Brauer groups and precise Hasse obstructions on normic bundles","Normic bundles: prescribe Brauer groups and minimal Hasse obstructions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001252,"raw_usage":{"total_tokens":5093,"prompt_tokens":868,"completion_tokens":4225,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":4143}},"tokens_in":484,"tokens_out":4225,"duration_ms":26107,"temperature":1.0,"reasoning_tokens":4143,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:20:13.303051+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Specialize to $k=\\mathbb{Q}$, $n=2$, $B=\\mathbb{Z}/2$, and $B_0=B$, choose $a$ and $P'$ as in Proposition 5.6, and choose $h$ as in Proposition 2.10. Compute directly the unramified Brauer group and the local images of the pulled-back model $X=X'\\times_{\\mathbb{P}^1,h}\\mathbb{P}^1$; if either differs from $B$ or from the prescribed set $\\Lambda$, the model assertion on which Theorem 5.1 rests is false.","supporting_citations":[],"review_version":1}