{"id":"aa09f102-4b42-428f-827c-ca6e96ebd1be","arxiv_id":"2508.21705","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":9.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The Iarrobino scheme compactifying oriented Gorenstein points is constructed for all quasi-projective schemes and is proven smooth over smooth curves, with applications to Hilbert scheme deformation theory and characteristic numbers.","lead":"This paper constructs a new moduli space, the Iarrobino scheme, which completes the locus of oriented Gorenstein zero-dimensional subschemes of a scheme X using self-dual filtrations and completed quadrics. It proves the space is smooth and flatly related to the Hilbert scheme when X is a smooth curve, with applications to deformation theory and enumerative geometry.","discovery_kind":"paradigm_shift","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The integral-fibre claim in Theorem 4.18 is unsupported: irreducibility of fibres is asserted but never proved, and the connectedness proof of Theorem 4.14 defers a key step to the reader.","rationale":"The reader's conditional verdict is well calibrated, but the weakest assumption is not the one I would single out. The lifting step in Theorem 4.14 is a genuine presentational gap, yet it is likely fillable by standard formal smoothness of CQ(fM') and of the compatible bundle over it. The more load-bearing omission is the integrality/irreducibility of the fibres in Theorem 4.18. The proof never even addresses irreducibility of a fibre; the reducedness argument from one smooth point is insufficient; and the total-space irreducibility depends on a connectedness proof whose final step is explicitly left to the reader. The construction of Iard(X) itself is coherent, the external inputs (TK88, Tyrrell) are standard, and there is independent support from examples, so a reject is not warranted; the paper needs either a completed proof or a weakened statement. This matches the existing CONDITIONAL verdict, so no change to the reader's verdict is needed.","tokens_in":35173,"tokens_out":27980,"duration_ms":288031,"concrete_test":"Compute the fibre F_3 = tau_{A1}^{-1}(Spec(k[x]/x^3)) inside Iar_3(A1) in Tyrrell coordinates, as in Example 4.19, and calculate its reduced ideal and primary decomposition. If F_3 is reducible, Theorem 4.18 is false. If F_3 is irreducible, expand the proof of Theorem 4.14's third step to give an explicit degeneration from (4.17) to M0, and supply the missing fibre-irreducibility argument; without that expansion, the theorem remains unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.18 states that tau_C has integral fibres and (tau_C)_*O = O. Its proof says 'We will show below that the fibres ... are reduced, irreducible of dimension d-1', but the displayed argument only establishes the dimension bound d-1 and then argues reducedness from the existence of a smooth unbroken point. A smooth point in one component does not give generic reducedness of a possibly reducible fibre. No argument for irreducibility of a fibre is supplied. The proof of Theorem 4.14, which is the source of irreducibility of Iard(C), also ends by delegating a required deformation ('connect any module of the form (4.17) ... to the module M0') with 'This is an elementary construction, which we leave to the reader', and Example 4.3 similarly says the special fibre is irreducible and 'A willing reader can do this now by hand.' These are not optional details: they are exactly the steps needed to rule out extra components of Iard(C) or of the fibres, and they underpin both the irreducibility statement and the Stein-factorization conclusion (tau_C)_*O = O.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces, for any quasi-projective k-scheme X (char k != 2) and d >= 0, a scheme Iard(X) that parameterizes flags X = Z0 ? Z1 ? ... of zero-dimensional subschemes of degree d together with symmetric isomorphisms (I_{Z_{i+1}}/I_{Z_i})^vee -> I_{Z_{i+1}}/I_{Z_i}, and a projective forgetful morphism tau: Iard(X) -> Hilbd(X). The construction is made through the variety of completed quadrics and new compatible and anticompatible vector bundles on it. The paper proves the scheme exists and is quasi-projective, gives a concrete description of its k-points, defines Quot, stack, and commuting-matrix analogues, and then studies the curve case: Theorem 4.14 claims CQuotd(E) is smooth and connected of dimension d(1 + rk E) - 1, and Theorem 4.18 claims Iard(C) is smooth, irreducible, of dimension 2d - 1, with flat projective integral fibres over Hilbd(C) satisfying (tau_C)_* O = O. Applications to deformation theory of Hilbert schemes, characteristic numbers of algebras, and enumerative geometry are discussed.","tokens_in":35537,"tokens_out":5540,"duration_ms":52893,"significance":"If the main theorems hold, this is a substantial new construction: it gives a natural compactification of the oriented Gorenstein locus, a self-dual analogue of the Hilbert and Quot schemes, and a new bridge between Hilbert schemes and the completed quadrics used in recent enumerative work. The explicit pointwise description of Iard(X) and the self-contained treatment of completed quadrics and of the (anti)compatible bundles are clear strengths. The paper also convincingly situates the construction relative to existing literature and to ongoing work. However, the central new geometric statements for curves are not fully proved as written: the smoothness and connectedness proof of Theorem 4.14 contains two unproved lifting/extension steps, and the irreducibility and integrality of the fibres in Theorem 4.18 are asserted rather than established. The significance of the paper is therefore conditional on repairing these arguments.","major_comments":[{"comment":"The infinitesimal lifting argument is plausible, but two steps are load-bearing and unproved. First, the sentence 'The element [eq•] is a map Spec(A) -> CQ(fM), -> CQ(fM')' assumes a natural morphism from the completed quadrics of a quotient module fM to the completed quadrics of a free extension fM'; no such inclusion is proved, and it is not automatic for completed quadrics on quotients. Second, the statement that the compatible bundle C is smooth over CQ(fM') does not by itself imply that a section Spec(A) -> C over CQ(fM')|Spec(A) extends to Spec(B) -> C, because the base map Spec(A) -> CQ(fM') must simultaneously be lifted to Spec(B) and the section must be lifted compatibly. These two points are exactly what is needed to conclude that the deformation extends, so Theorem 4.14 is not justified as written.","section":"Section 4.4, Theorem 4.14 (smoothness)"},{"comment":"The connectedness proof ends by reducing to torus-fixed quotients of the form (4.17) and then says 'connect any module of the form (4.17) ... to the module M0. This is an elementary construction, which we leave to the reader.' Example 4.3 similarly says 'A willing reader can do this now by hand' for the irreducibility of the compatible locus over k[x]/(x^d). These are not cosmetic exercises: without an explicit deformation argument one cannot conclude that the open locus V is dense or that there are no extra components, and the claim that CQuotd(E) is connected remains incomplete.","section":"Section 4.4, Theorem 4.14 (connectedness)"},{"comment":"The assertion that every fibre tau_C^{-1}([Z]) is irreducible is never proved. The displayed argument only establishes the dimension bound and then deduces reducedness from the existence of a smooth unbroken point; a smooth point in one component of a possibly reducible fibre does not imply generic reducedness. The integrality of the fibres is used both for flatness via Miracle Flatness and for the Stein factorization conclusion (tau_C)_* O = O, so the statement of Theorem 4.18 is not supported by the proof as written.","section":"Section 4.4, Theorem 4.18"},{"comment":"The proof of independence of characteristic numbers uses flatness of tau_{A^1} via [Ful98, Section 10.2]. Since the flatness of tau_{A^1} is a consequence of Theorem 4.18, this application is conditional on the missing fibre-integrality proof. The paper should either prove the needed flatness directly for the universal family over Hilbd(A^1) or state explicitly that the result is conditional on Theorem 4.18 being completed.","section":"Section 4.4.1, Proposition 4.21"}],"minor_comments":[{"comment":"The word 'bijectively' appears throughout; this should be replaced by 'bijectively'.","section":"Throughout, e.g. Abstract and Proposition 4.7"},{"comment":"The manuscript uses 'anticompactible' once; this is a typo for 'anticompatible'.","section":"Section 3.4, after Definition 3.6"},{"comment":"The check that (3.17) is the limit of Lambda^bullet q(t) is left to the reader; a short justification would improve readability, though this is not load-bearing.","section":"Lemma 3.18"},{"comment":"The notation A^{d-1} is used both for affine space and for the base field; this is potentially confusing but harmless.","section":"Proposition 3.26 and Example 3.31"},{"comment":"The reference [JPS25] is cited as arXiv:19587, which appears to be an incomplete identifier; the full arXiv number should be given.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The construction is novel and the overall architecture of the paper is convincing. The main issue is that the proof of the central curve theorem contains real gaps, especially the lifting step in the smoothness proof of Theorem 4.14 and the missing irreducibility argument in Theorem 4.18. These gaps appear fixable within the scope of the paper, and I would encourage a revised version rather than a rejection. The paper's reliance on [JRS25] for some applications is acceptable, but the central theorem should not depend on future work."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper introduces a genuinely new moduli object, and the main construction is careful and well motivated, but the curve-case theorems are not fully proved as written. The gaps are concentrated in the connectedness argument and in the integral-fibre claim, and both are load-bearing. What is actually new: Iard(X), CQuotd(E), and the stack analogues are absent from prior literature. The compatible and anticompatible bundles are new, and the reinterpretation of Iarrobino's symmetric decomposition as a torus limit is valuable. The construction is explicit, the commuting symmetric matrices connection is a bonus, and the citation pattern is fine. On Theorem 4.14, the lifting step is compressed but fixable: smoothness of CQ(fM') over B gives the lift, and the compatible bundle then gives the operator lift. The real soft spots are connectedness and fibres. The proof that V is dense ends with a step left to the reader: connecting any module of form (4.17) to M0, which is exactly what rules out extra components. Then Theorem 4.18 asserts fibres reduced and irreducible but only proves dimension and a smooth unbroken point; irreducibility of fibres is not proved. These are not cosmetic. The central framework is coherent, the missing arguments are likely fillable. This should go to a serious referee, not desk-rejected, but the referee should demand a complete connectedness argument and a direct proof of fibre irreducibility.","headline":"New moduli construction worth taking seriously; the main theorems are plausible but the printed proofs leave two load-bearing gaps that a revision should fill.","tokens_in":752,"tokens_out":1520,"would_cite":true,"duration_ms":55022,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C05","14D22","14M27","13H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The Iarrobino scheme is a self-dual Hilbert scheme of points, and over any smooth curve it is smooth of dimension $2d-1$.","keywords":["Iarrobino scheme","Hilbert scheme of points","self-dual modules","Gorenstein schemes","completed quadrics","Quot scheme of points","symmetric decomposition","moduli spaces"],"falsifier":"For $C = \\mathbb{A}^1$ and $E = O^{\\oplus r}$, compute the complete local ring at a point of $\\mathrm{CQuot}^d(E)$ corresponding to a split module with a rank-one first quadric; if its tangent space has dimension larger than $d(1+r)-1$, the infinitesimal lifting assertion in the proof of Theorem 4.14 is false.","tokens_in":1880,"feed_emoji":"📐","tokens_out":11323,"duration_ms":150884,"temperature":0.7,"pith_summary":"The paper introduces the Iarrobino scheme, a self-dual analogue of the Hilbert scheme of points. For a scheme $X$, it parameterises flags of zero-dimensional subschemes together with symmetric isomorphisms between successive quotient modules. This compactifies the locus of oriented Gorenstein subschemes, where the extra data is a trivialisation of the dualising sheaf. The main result is that for a smooth curve $C$, the space $\\operatorname{Iar}_d(C)$ is smooth, irreducible, of dimension $2d-1$, and the forgetful map $\\tau_C$ is flat and projective with integral fibres of dimension $d-1$.","feed_headline":"Self-dual Hilbert scheme is smooth on curves","feed_subtitle":"A new moduli space compactifies oriented Gorenstein subschemes; over a curve it is flat with integral fibres.","key_machinery":"The load-bearing object is the variety of completed quadrics $CQ(V)$, the closure of the space of full-rank quadrics inside a product of projective spaces of exterior powers. A point of $CQ(V)$ is a broken quadric: a sequence of symmetric maps $q_0: V^\\vee\\to V$, then $q_1: \\ker q_0 \\to \\operatorname{coker} q_0$, and so on, ending with a full-rank quadric. The paper defines compatible and anticompatible bundles on $CQ(V)$, whose fibres are endomorphisms preserving the associated flag and acting symmetrically or antisymmetrically on each subquotient. The Iarrobino scheme is then the compatibility locus of the universal family over the Hilbert scheme with the compatible bundle, and the smoothness results for curves are proved using the smoothness of $CQ(V)$, Tyrrell's explicit affine patches, and an infinitesimal lifting argument.","core_discovery":"For every quasi-projective $k$-scheme $X$ with $\\operatorname{char} k \\neq 2$ and every integer $d$, the paper constructs a quasi-projective scheme $\\operatorname{Iar}_d(X)$ with a projective morphism $\\tau_X:\\operatorname{Iar}_d(X)\\to \\mathrm{Hilb}_d(X)$. Its $k$-points are flags $X \\supsetneq Z_0 \\supsetneq \\cdots$ of zero-dimensional subschemes of degree $d$, together with symmetric $O_X$-module isomorphisms $q_i: (I_{Z_{i+1}}/I_{Z_i})^\\vee \\to I_{Z_{i+1}}/I_{Z_i}$. This compactifies the oriented Gorenstein locus, where there is only one term and $q_0$ trivialises the dualising sheaf. The central structural result is that for a smooth connected curve $C$, the scheme $\\operatorname{Iar}_d(C)$ is smooth and irreducible of dimension $2d-1$, the forgetful map $\\tau_C$ is flat and projective with integral fibres of dimension $d-1$, and $(\\tau_C)_*O_{\\operatorname{Iar}_d(C)} = O_{\\mathrm{Hilb}_d(C)}$.","pith_inferences":["If the compatibility-locus construction is as robust as it appears, the same smoothness argument might extend to other bases where completed quadrics are smooth, giving self-dual Quot schemes beyond the curve case without additional hypotheses.","The interpretation of Iarrobino's symmetric decomposition as a torus limit on $\\operatorname{Iar}_d(\\mathbb{A}^n)$ suggests that Bialynicki-Birula decompositions of the Iarrobino scheme could produce new numerical constraints on Hilbert functions of Gorenstein algebras.","A concrete testable direction is to compute intersection numbers on $\\operatorname{Iar}_d(X)$ for $X$ of higher dimension, where the fibres of $\\tau_X$ need not be irreducible; flatness would then be replaced by virtual structure, giving a different enumerative theory from the one on the usual Hilbert scheme."],"forward_implications":["Over a smooth curve $C$, $\\operatorname{Iar}_d(C)$ is smooth, irreducible of dimension $2d-1$, and $\\tau_C$ is flat and projective with integral fibres of dimension $d-1$.","For a smooth threefold $X$, a point of $\\operatorname{Iar}_d(X)$ whose intermediate subquotients are principal ideals detects smooth points of $\\mathrm{Hilb}_d(X)$, giving a new route toward understanding smoothness of Hilbert schemes of points on threefolds.","The characteristic numbers of the algebra $k[x]/(f)$ are independent of $f$, because the relevant subvariety is a fibre of the flat map $\\tau_{\\mathbb{A}^1}$ over the Hilbert scheme.","The Iarrobino scheme provides a modular interpretation of the varieties $X_V$ attached to finite algebras as closures of the unbroken part of the fibre of $\\tau$, explaining their lower semicontinuity.","The same construction yields self-dual analogues of the Quot scheme of points and of the stacks of coherent sheaves and finite algebras, so the formalism extends beyond Hilbert schemes."],"supporting_citations":[{"why":"Constructs the variety of completed quadrics and its functor of points, giving the projective model on which the Iarrobino scheme is built.","marker":"[TK88]"},{"why":"Supplies the explicit affine patches on $CQ(V)$ used to construct and verify the compatible and anticompatible bundles.","marker":"[Tyr56]"},{"why":"Establishes the closure description of $CQ(V)$ in the product of projective spaces of exterior powers.","marker":"[Lak87]"},{"why":"Provides the wonderful-compactification viewpoint that yields the anticompatible bundle as the kernel of the logarithmic tangent map.","marker":"[DCP83]"},{"why":"Introduces the symmetric decomposition of Hilbert functions, whose geometric limit is realised as a torus limit in the Iarrobino scheme.","marker":"[Iar94]"},{"why":"Gives the Bialynicki-Birula decomposition used to identify torus limits and to prove connectedness.","marker":"[BB73]"},{"why":"Buchsbaum-Eisenbud structure theorem is used to show that the unbroken Gorenstein locus is smooth in dimension at most three.","marker":"[BE77]"}],"fun_headline_variants":["Iarrobino scheme: self-dual Hilbert space is smooth on curves","New moduli space for oriented Gorenstein subschemes is smooth on curves","Compactifying Gorenstein subschemes: Iarrobino scheme is smooth for curves","Self-dual analogue of Hilbert scheme is smooth and flat on curves","Iarrobino scheme: a smooth compactification of Hilbert scheme"],"cache_read_input_tokens":38016,"weakest_assumption_plain":"The smoothness proof for the self-dual Quot scheme over a curve assumes that every compatible self-duality structure on an infinitesimal deformation can be lifted to the next infinitesimal order once the underlying module is freely extended; if that lifting fails, the smoothness claim does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Iarrobino scheme: self-dual Hilbert space is smooth on curves","New moduli space for oriented Gorenstein subschemes is smooth on curves","Compactifying Gorenstein subschemes: Iarrobino scheme is smooth for curves","Self-dual analogue of Hilbert scheme is smooth and flat on curves","Iarrobino scheme: a smooth compactification of Hilbert scheme"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000862,"raw_usage":{"total_tokens":3773,"prompt_tokens":1013,"completion_tokens":2760,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":2659}},"tokens_in":629,"tokens_out":2760,"duration_ms":17182,"temperature":1.0,"reasoning_tokens":2659,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:38:42.758488+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $C = \\mathbb{A}^1$ and $E = O^{\\oplus r}$, compute the complete local ring at a point of $\\mathrm{CQuot}^d(E)$ corresponding to a split module with a rank-one first quadric; if its tangent space has dimension larger than $d(1+r)-1$, the infinitesimal lifting assertion in the proof of Theorem 4.14 is false.","supporting_citations":[],"review_version":2}