{"id":"d0e83175-def2-430d-825e-b2456f85e529","arxiv_id":"2501.00105","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A conditional spectral sequence model for algebraic morphism spaces is announced, with an inconsistent stable-range formula and an off-by-two conflict between theorem and proof.","lead":"This paper claims a spectral sequence describing the cohomology of the space of algebraic maps between projective varieties using configuration spaces of the source, a higher-dimensional analogue of Bendersky and Gitler's topological theorem. The statement is conditional on strong, unproved Leray-Hirsch hypotheses for auxiliary schemes, and the paper's explicit formula for the stable range does not match its own cited estimate.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.0.1(3) mis-solves the Angehrn-Siu inequality and overclaims a sufficient condition as an exact equality; the stated r(d) formula is false (P^2, d=O(1) gives r=2 vs formula's 1).","rationale":"The paper's central claim is the configuration-space spectral sequence for Mor_d(X,Y). The reader's weakest_assumption, the §2.5.4 Leray-Hirsch condition, is indeed a serious limitation, and Remark 2.5.5 concedes that no general sufficient conditions are known for arbitrary Y. That alone would make the theorem conditional and of narrow applicability. However, the most decisive problem is the explicit r(d) formula: the proof of statement 3 is a one-line citation to Angehrn-Siu, but the formula is both algebraically mis-derived and logically overclaimed, since a sufficient condition cannot determine the exact maximal r(d). The concrete P^2 example shows the stated equality is false. The reader also noted the p-range mismatch in the proof of statement 1 and the exact-equality overclaim; both compound the issue. Because the theorem as stated is false, REJECT is appropriate. The spectral-sequence machinery may be salvageable after correcting the bound, stating the §2.5.4 hypotheses explicitly, and adjusting the range, but the announced result cannot be accepted in its current form.","tokens_in":16359,"tokens_out":18720,"duration_ms":175188,"concrete_test":"Compute r(d) for X = P^2 and d = O(1). Then δ = O(4), and the minimum of (δ^k·[W])^{1/k} over [W] ∈ CH_k(X), k = 1,2, is 4 (attained both by a line and by X itself). The formula in (1.0.5) gives floor((2·4 − 4 + 2 − 1)/2) − 1 = 1. Directly, O(1) separates any two points of P^2 (the restriction map H^0(O(1)) → H^0(O_Z) is surjective for |Z| = 2) but does not separate three collinear points, so r(d) = 2. The mismatch disproves the claimed equality in Theorem 1.0.1(3).","verdict_should_be":"REJECT","load_bearing_attack":"Statement 3 of Theorem 1.0.1 asserts an exact equality for r(d) derived from the Angehrn-Siu criterion (Theorem 2.2.1 and (2.3.1)). That criterion is only a sufficient condition: if (δ^k·[W])^{1/k} > (n/2)(n+2r−1) for every irreducible subvariety W of dimension k, then K_X+δ separates r points. Solving for the largest integer r gives r < (1/n)·min_{[W],k}(δ^k·[W])^{1/k} − n/2 + 1/2, with strict inequality. The expression displayed in (1.0.5) is different: it drops the division by n and, more seriously, converts a sufficient lower bound into an exact equality for the actual r(d). This is not a cosmetic slip. For X = P^2 and d = O(1), one has δ = d − K_X = O(4), d is acyclic, and the formula in (1.0.5) evaluates to 1; but O(1) on P^2 separates any two distinct points (evaluation at two points is surjective) and fails to separate three collinear points, so the true value is r(d) = 2. Thus the theorem's advertised equality is false, and the quantitative ranges that depend on r(d) in statements 1 and 2 are not supported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a configuration space model for the cohomology of the moduli space Mor_d(X,Y) of algebraic maps between smooth projective varieties. For acyclic d and a polarized target Y, it constructs a proper hypercover of the discriminant locus and, under a Q-Leray-Hirsch assumption, derives a first-quadrant spectral sequence whose E_1-term is expressed in terms of cohomology of configuration spaces of X, the Picard variety, and auxiliary schemes Y(D_p). For Y = P^N it claims the sequence collapses at E_2 in a range, and it gives an explicit intersection-theoretic formula for the stable range r(d) based on the Angehrn-Siu criterion. The paper is a continuation of the author's earlier work on curves and uses the language of derived infinity-categories of constructible sheaves.","tokens_in":16773,"tokens_out":14704,"duration_ms":126789,"significance":"If the theorem were correct, it would be a striking algebro-geometric analogue of Bendersky-Gitler's result, with additional Galois/MHS structure. The hypercover construction is natural and the use of symmetric simplicial descent is elegant. However, the stated formula for r(d) is false, the range in the spectral sequence is inconsistent between theorem and proof, and the general case rests on an unproven Leray-Hirsch condition with no known verifying examples beyond projective space. The paper therefore does not currently establish its main claims.","major_comments":[{"comment":"The displayed formula for r(d) is obtained by solving the Angehrn-Siu inequality (2.2.1) incorrectly, and it also converts a sufficient condition into an exact equality. From (2.2.1), (δ^k·[W])^{1/k} > (n/2)(n+2r−1), the largest integer r satisfies r < (δ^k·[W])^{1/k}/n − (n−1)/2, not r = floor(2(δ^k·[W])^{1/k} − n/2 + (n−1)/2) − 1. Moreover, Angehrn-Siu only guarantees separation, so the maximum number of separated points can be larger than the bound. A concrete failure is X = P^1, d = O(1): here n = 1, δ = d − K_X = O(3), and (1.0.5) evaluates to 4, while O(1) separates exactly 2 points (H^0(O(1)) has dimension 2, and evaluation at 3 points cannot be surjective). Thus Theorem 1.0.1(3) is false as stated, and statements 1 and 2 inherit an incorrect quantitative range.","section":"§2.3.1, Eq. (2.3.1) and Theorem 1.0.1(3), Eq. (1.0.5)"},{"comment":"The theorem states that the E_1 description (1.0.3) holds for all p ≤ r(d)+1, but the proof in §4.1, equation (4.1.9), says 'for all p ≤ r(d)−1'. Later, (4.2.1) says 'for all p ≤ r(d)' and (4.2.2) uses p ≤ r(d)+1. These three ranges cannot all be correct. If the intended range is r(d)+1, the proof must justify the two additional degrees; if it is r(d)−1, the theorem overreaches. This is a load-bearing point because the range is the main quantitative output of the configuration space model.","section":"Theorem 1.0.1(1) vs. §4.1.9, Eq. (4.1.9)"},{"comment":"The theorem's statement for arbitrary Y is conditional on an assumption that is not listed as a hypothesis. Paragraph 2.5.4 assumes that Mor_d(X,Y) is nonempty and that, for all r ≤ r(d), X_r(Y) is nonempty and satisfies Q-Leray-Hirsch on each stratum. Remark 2.5.5 states that no sufficient conditions for such a stratified fibre bundle structure are known for a reasonably large class of Y. Consequently, outside the case Y = P^N (where Lemma 2.4.7 provides the structure), the theorem appears to have no verified instances, and the general statement is not established. The theorem should state this assumption explicitly and either provide examples or restrict to cases where the condition can be verified.","section":"§2.5.4 and Theorem 1.0.1"},{"comment":"The proof of Lemma 3.1.5 is not self-contained: the abelian case is quoted from [Ban24, Lemma 2.7] and the infinity-categorical adaptation is only sketched via a spectral sequence in (3.1.4)–(3.1.5). Since this lemma is essential for replacing the simplicial cochain complex by its sign-twisted invariants in (4.1.5), the reader cannot verify a key step without reconstructing the argument from [FL91] and [Lur17]. Please provide a complete proof or a precise reference that states the infinity-categorical version.","section":"Lemma 3.1.5, §3.1.4–3.1.5"}],"minor_comments":[{"comment":"The projection pr23 is defined twice with different meanings: in (2.4.10) it denotes projection to the first two factors, which should be called pr12. The same mislabeling occurs in (2.4.18)–(2.4.20). This makes the proof of Lemma 2.4.4 unnecessarily hard to follow.","section":"§2.4.4, Eqs. (2.4.8)–(2.4.10) and (2.4.18)–(2.4.20)"},{"comment":"The claim that the symmetric group S_{r+1} acts freely on X_r is false when some of the x_i coincide, since such tuples have nontrivial stabilizers. The Delta^S-structure does not require freeness, so the wording should be corrected.","section":"§3.1.1"},{"comment":"The abstract contains grammatical slips: 'admit a configuration space model' should be 'admits a configuration space model', and 'should be a thought of' should be 'should be thought of'.","section":"Abstract and Introduction"},{"comment":"The condition 'g(s_0,...,s_N) = 0' is written as if g were a single polynomial, but in the setup g = (g_1,...,g_m) is a set of generators. The equations should state g_i(s_0,...,s_N) = 0 for all i.","section":"Eq. (2.5.1) and surrounding text"},{"comment":"The reference [Har77] is dated '977' and should read '1977'. Also, several arXiv references (e.g., [Aum24], [Ban22], [Men21]) would benefit from version numbers and the year of the arXiv posting.","section":"References"}],"recommendation":"reject","confidential_remarks":"The manuscript leans heavily on the author's own recent papers, especially [Ban24] and the arXiv preprint [Ban22], for the symmetric simplicial machinery and for an abelian version of Lemma 3.1.5. The referee did not find this circular in the strict sense, but the reliance is substantial. Given the false formula in Theorem 1.0.1(3) and the inconsistency in the spectral sequence range, I recommend rejection; a major revision would need to correct these points and clarify the status of the 2.5.4 assumption before the paper could be considered further."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe headline: the main theorem as stated is false, but the machinery behind it is new and possibly salvageable. The paper's stable range formula (1.0.5) mis-solves the Angehrn-Siu inequality, and that error propagates into statements 1 and 2.\n\nWhat is genuinely new: Banerjee gives a higher-dimensional algebro-geometric configuration space model for algebraic maps, extending her curve case and Bendersky-Gitler. The construction of a proper hypercover over the discriminant locus, the use of the symmetric simplicial category and ∞-categorical descent, and the resulting spectral sequence with Galois and Hodge structures is a real idea. For Y = P^N the E1 term is explicit and the geometry of the X_r as projectivized stratified vector bundles is concrete. That part deserves attention.\n\nThe soft spots are serious. Solving (2.2.1) for r gives r < (1/n)(δ^k·[W])^{1/k} − n/2 + 1/2, not 2(δ^k·[W])^{1/k} − n/2 + (n−1)/2. The missing division by n is not cosmetic. On X = P^2 with d = O(1), the true r(d) is 2 (O(1) separates two points but not three collinear ones), whereas (1.0.5) evaluates to 6. So the advertised equality in Theorem 1.0.1(3) is false. Moreover, Angehrn-Siu is a sufficient criterion, so even a correct bound would be a lower bound on r(d), not an exact equality.\n\nThere's also an indexing inconsistency: the theorem states the E1 description for p ≤ r(d)+1, but the proof of (4.1.9) only claims it for p ≤ r(d)−1, and then (4.2.2) switches back to p ≤ r(d)+1. That needs to be sorted out.\n\nFinally, the blanket assumption in §2.5.4 — that every X_r(Y) is nonempty and satisfies Q-Leray-Hirsch on each stratum — is exactly what makes the arbitrary-Y result work, and the paper concedes no general sufficient conditions are known. So the theorem is really about P^N plus whatever sporadic cases satisfy that hypothesis.\n\nWho should read this? People working on stable cohomology of moduli of maps, configuration space models, and the intersection-theoretic approach to point separation. The paper is not ready for publication, but it is not a waste of time: the hypercover/∆S framework is worth a serious look, and a referee could help the author fix the bound and the indexing.\n\nMy recommendation: send it to review if the author is willing to address the formula error and the range-of-p issue. A serious referee is warranted because the core construction is novel and the flaws are localized, even though the theorem as written is wrong.","headline":"The stable range formula in Theorem 1.0.1 mis-solves Angehrn-Siu, so the theorem as written is false, but the hypercover/∆S spectral sequence idea is new and likely repairable.","tokens_in":17249,"tokens_out":4987,"would_cite":false,"duration_ms":44056,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D22","14C30","14F08","14F20","14C17"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under acyclicity and point-separation hypotheses, the cohomology of the algebraic map space $\\operatorname{Mor}_d(X,Y)$ is computed by a spectral sequence built from configuration spaces of $X$, the Picard variety, and auxiliary schemes…","keywords":["configuration space model","algebraic map spaces","spectral sequence","mixed Hodge structures","Galois representations","homological stability","hypercovers","point-separating line bundles"],"falsifier":"Compute the $E_1$ page and $H_c^*(\\operatorname{Mor}_d(X,Y);\\mathbb{Q})$ for an explicit pair satisfying the hypotheses of Theorem 1.0.1—say $X=\\mathbb{P}^1$, $Y=\\mathbb{P}^2$ with a fixed degree $d$, checked over a finite field by comparing Frobenius traces on both sides. A disagreement at any bidegree with $p\\le r(d)+1$ would falsify the configuration-space model.","tokens_in":16163,"feed_emoji":"🧩","tokens_out":15470,"duration_ms":140447,"temperature":0.7,"pith_summary":"The paper proves that, when the degree $d$ is acyclic and separates $r(d)$ points, the cohomology of the moduli space $\\operatorname{Mor}_d(X,Y)$ of algebraic maps between smooth projective varieties is governed by a first-quadrant spectral sequence whose $E_1$ page is made from the sign-twisted symmetric invariants of the cohomology of the configuration space $\\mathring{X}^p$, the cohomology of the Picard variety $\\operatorname{Pic}^d(X)$, and compactly supported cohomology of auxiliary schemes attached to $Y$. This is an algebro-geometric analogue of the classical configuration-space model for continuous function spaces, and it is stronger than the topological analogue in that every term and differential carries Galois representations and mixed Hodge structures. For maps to projective space the sequence collapses at $E_2$ in a stable range, and the length of that range is an explicit intersection-theoretic number computed from $X$ alone. A reader should care because it reduces a rigid and difficult object—the space of algebraic maps—to configuration spaces and intersection theory, where the structure is explicit.","feed_headline":"Algebraic map cohomology comes from configuration spaces","feed_subtitle":"A configuration-space spectral sequence computes algebraic-map cohomology, Hodge structures included.","key_machinery":"The load-bearing object is a proper hypercover $\\pi_\\bullet:X_\\bullet(Y)\\to Z_d(X,Y)$ of the discriminant locus $Z_d(X,Y)$ of the natural compactification of $\\operatorname{Mor}_d(X,Y)$, together with the symmetric simplicial category $\\Delta^S$, a variant of the simplex category in which each level carries symmetric-group actions. The hypercover is built by recording, along with a tuple of sections defining a map, the common zero loci of those sections; for $Y=\\mathbb{P}^N$ its levels $X_r$ are projectivizations of stratified vector bundles over $X^{r+1}\\times \\operatorname{Pic}^d(X)$, which makes them stratified fiber bundles satisfying Q-Leray–Hirsch. Cohomological descent identifies the constant sheaf on $Z_d(X,Y)$ with the total complex of the hypercover, and passage through $\\Delta^S$ with sign-twisted $S_p$-invariants transforms this into a Koszul-type cochain complex, equation (4.1.5). The key formal identity is Lemma 3.1.5, which identifies the alternating-face complex of a $\\Delta^S$-sheaf with the sign-twisted symmetric-invariant complex; this is what turns the hypercover into the configuration-space $E_1$ page.","core_discovery":"The paper's central claim is Theorem 1.0.1. Let $X$ be a smooth projective $n$-fold, $(Y,\\Upsilon)$ a polarized smooth projective variety, and $d\\in N^1(X)$ acyclic and separating $r(d)$ points. Then there is a first-quadrant spectral sequence of Galois representations and mixed Hodge structures $E_1^{p,q}\\Rightarrow H_c^{p+q}(\\operatorname{Mor}_d(X,Y);\\mathbb{Q})$, and for $p\\le r(d)+1$ the page is $$$E_1^{{p,*}}$ = \\big((H^*(\\mathring{X}^p;\\mathbb{Q})\\otimes \\operatorname{sgn}_{S_p})^{S_p}\\big)\\otimes H^*(\\operatorname{Pic}^d(X);\\mathbb{Q})\\otimes H_c^*(Y(D_{p-1});\\mathbb{Q}),$$ where $\\mathring{X}^p$ is $X^p$ with all diagonals removed and $Y(D_{p-1})$ are auxiliary schemes defined in §2.5.4. When $Y=\\mathbb{P}^N$, the sequence collapses at $E_2$ for $0\\le p\\le r(d)+1$ and $\\dim\\operatorname{Mor}_d(X,Y)-r(d)\\le q\\le \\dim\\operatorname{Mor}_d(X,Y)$, yielding homological stability in that range. If $\\delta=d-c_1(K_X)$ is ample, the stable bound satisfies $$r(d)=\\left\\lfloor\\min_{[W]\\in CH^k(X),\\,1\\le k\\le n}\\left(2(\\delta^k.[W])^{1/k}-\\frac{n}{2}+\\frac{n-1}{2}\\right)\\right\\rfloor-1.$$ Thus the cohomology of algebraic map spaces is controlled by the configuration spaces of the domain, the Picard variety, and auxiliary moduli of $Y$, with arithmetic and Hodge structure intact.","pith_inferences":["If the model holds, the stable cohomology of $\\operatorname{Mor}_d(X,\\mathbb{P}^N)$, viewed as a functor of the polarization degree $d$, should exhibit representation stability in the symmetric-group direction; the paper does not explore this consequence.","Because all steps are algebraic, one can test the theorem arithmetically: over a finite field, the trace of Frobenius on $H_c^*(\\operatorname{Mor}_d(X,Y))$ should equal the alternating sum of traces on the $E_1$ page in the claimed range, a check accessible for low-dimensional $X$ and $Y$ without computing the whole spectral sequence.","A natural first test beyond projective space is $X=\\mathbb{P}^1$ and $Y$ a smooth low-degree hypersurface, where point-counting estimates for mapping spaces are available; the paper itself notes that no general condition guaranteeing the §2.5.4 hypothesis is known, so such a check would decide whether the model has a genuinely wider scope."],"forward_implications":["For any polarized target satisfying the §2.5.4 assumptions, the Galois representation and mixed Hodge structure on $H_c^*(\\operatorname{Mor}_d(X,Y);\\mathbb{Q})$ is packaged by the configuration-space $E_1$ page, so those structures on the map space are determined by the corresponding structures on $\\mathring{X}^p$, $\\operatorname{Pic}^d(X)$, and $Y(D_{p-1})$.","For $Y=\\mathbb{P}^N$, the collapse at $E_2$ in the stated range gives explicit stable cohomology of $\\operatorname{Mor}_d(X,\\mathbb{P}^N)$ from $H^*(X)$, $H^*(\\operatorname{Pic}^d(X))$, and projective-space factors, and implies homological stability near the top of the cohomological range.","The stable range $r(d)$ is an intersection-theoretic invariant of $X$: when $\\delta=d-c_1(K_X)$ is ample it is given by (1.0.5), so the more positive $\\delta$ is, the larger the range in which the configuration-space model computes the full cohomology.","The same hypercover-to-Koszul mechanism, with separation of points replaced by jet-ampleness, also applies to spaces of sections of vector bundles, reproducing the cohomological results of the cited work [DH24] by this method."],"supporting_citations":[{"why":"Supplies the topological function-space theorem whose configuration-space model this paper carries over to algebraic maps.","marker":"[BG91]"},{"why":"Provides the symmetric simplicial hypercover framework and Lemma 2.7 that the proof's descent-to-invariants step adapts.","marker":"[Ban24]"},{"why":"Gives the point-separation estimate used to derive the explicit stable bound $r(d)$ in (1.0.5).","marker":"[AS95]"},{"why":"Supplies the symmetric simplicial group results, especially Theorem 6.9 and Corollary 6.17, used to identify sign-twisted symmetric invariants with ordinary cohomology of the hypercover.","marker":"[FL91]"},{"why":"Sets up the derived $\\infty$-category of constructible sheaves with six-functor formalism in which cohomological descent and the Koszul complex are formulated.","marker":"[GL19]"},{"why":"Provides the enhanced six-operations and base-change formalism used for proper descent of the hypercover.","marker":"[LZ24]"},{"why":"Supplies the standard algebraic-geometry facts (flat base change, properness, projectivization) used to build the spaces $X_r$ and prove the hypercover is proper.","marker":"[Har77]"},{"why":"Supplies the earlier curve-domain implementation and the moduli-of-maps perspective on which the higher-dimensional extension is modeled.","marker":"[Ban22]"}],"fun_headline_variants":["Spectral sequence builds algebraic map cohomology from configurations","Configuration spaces compute cohomology of algebraic maps","Algebraic map cohomology via configuration spaces","Hodge-theoretic spectral sequence for algebraic map spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, for every $r\\le r(d)$, the auxiliary schemes $X_r(Y)$ are nonempty and have Q-Leray–Hirsch cohomology over each stratum of $X^{r+1}\\times\\operatorname{Pic}^d(X)$ (so that the $E_1$ page factors as a product), and that $\\operatorname{Mor}_d(X,Y)$ itself is nonempty; the paper states that no general sufficient condition for this is known for a reasonably large class of targets.","fun_headline_variants_meta":{"raw":{"variants":["Spectral sequence builds algebraic map cohomology from configurations","Configuration spaces compute cohomology of algebraic maps","Algebraic map cohomology via configuration spaces","Hodge-theoretic spectral sequence for algebraic map spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00022,"raw_usage":{"total_tokens":1463,"prompt_tokens":979,"completion_tokens":484,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":422}},"tokens_in":595,"tokens_out":484,"duration_ms":4802,"temperature":1.0,"reasoning_tokens":422,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:01:12.142169+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $E_1$ page and $H_c^*(\\operatorname{Mor}_d(X,Y);\\mathbb{Q})$ for an explicit pair satisfying the hypotheses of Theorem 1.0.1—say $X=\\mathbb{P}^1$, $Y=\\mathbb{P}^2$ with a fixed degree $d$, checked over a finite field by comparing Frobenius traces on both sides. A disagreement at any bidegree with $p\\le r(d)+1$ would falsify the configuration-space model.","supporting_citations":[],"review_version":1}