{"id":"7afd82d5-783a-4923-9be9-92583af19a9b","arxiv_id":"1908.08260","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The symplectic leaves of the Tyurin-Bottacin Poisson structure on stable pure dimension one sheaves on a Poisson surface are the fibers of the map sending a sheaf to the intersection of its support curve with the degeneracy divisor.","lead":"This paper determines the symplectic leaves of a natural Poisson structure on the moduli space of stable one-dimensional sheaves on a Poisson surface: they are exactly the fibers of the map that records how the support curve of the sheaf meets the divisor where the Poisson structure degenerates. The result transfers a known leaf description from Higgs bundles to sheaf moduli, giving a complete foliation picture.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1 proves tangent-space equality but not connectedness of fibers; without connectedness the statement 'fibers are symplectic leaves' is not justified.","rationale":"The reader's CONDITIONAL verdict is essentially right. The main mathematical identification T_E phi0 = T0 is standard and convincingly argued via the commutative diagram in Proposition 4.2 and the tangent sequence for f0. I do not see a fatal flaw in the deformation-theoretic core. The two concerns raised by the reader are of different weight. The false H^0 vanishing statement is real but does not affect the intended case: for a stable sheaf with smooth irreducible support, the anti-canonical restriction has negative degree and the vanishing holds; disconnected smooth supports would force the sheaf to be decomposable and hence not stable. The more serious issue is connectedness: tangent-space equality alone identifies each connected component of a fiber as a symplectic leaf, but the theorem asserts the whole fiber is one leaf. The proof never addresses whether fibers of g, or of the relative Picard map over them, are connected. This is a gap in the statement, not in the local computation, and it is probably repairable by standard torsor/projective-bundle arguments. I therefore keep the reader's CONDITIONAL verdict unchanged, with the caveat that the paper should either prove connectedness or restate Theorem 5.1 in terms of connected components of fibers.","tokens_in":7160,"tokens_out":55278,"duration_ms":582643,"concrete_test":"Verify connectedness of the fibers of g in a nontrivial example with q(S)>0: take S=P^1 x E, D=2f (two fibers of p), and C=2F+f, so d'=4. For a general Z in Sym^4(D), compute g^{-1}(Z) explicitly and check whether it is connected (it should be a projective bundle over a connected open subset of Pic^0(S)). If a fiber with two or more connected components can be exhibited, then Theorem 5.1's wording 'fibers are symplectic leaves' fails; if all such fibers are connected, the missing connectivity argument should be added to the proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 5.1 shows, pointwise, that the tangent space to the fiber of phi0 equals the image T0 of the Poisson map sigma. This is the main computation, and it appears correct. However, the theorem's conclusion that the fibers themselves are symplectic leaves requires each fiber to be a connected integral submanifold of the symplectic distribution. The paper does not prove connectedness of the fibers of g: Q0 -> Sym^{d'}(D), nor of the composed map phi0. Under the standard definition, a symplectic leaf is maximal and connected; a disconnected fiber would split into several leaves, and the statement 'fibers are symplectic leaves' would be false as written even though every tangent-space identification in the proof holds. The missing connectivity step is therefore load-bearing for the final formulation of the theorem, although it is likely fixable: the fiber of phi0 over Z is a relative Picard space over the fiber of g, and one would need to show that g^{-1}(Z) is connected, e.g., by realizing it as a projective bundle over a connected piece of Pic^0(S).\n\nThe overbroad claim in Section 5 that H^0(C,K_S|_C)=0 for every effective curve with C·D != 0 is indeed false for reducible or disconnected curves with a component disjoint from D. For the sheaves actually in M'_0, stability forces the support to be irreducible smooth, so the needed vanishing holds. This is a genuine wording error but not the main threat to Theorem 5.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the moduli space M_H(S,P) of H-stable pure dimension-one sheaves on a smooth complex projective Poisson surface (S,s), where s is a nonzero section of -K_S with degeneracy divisor D. The Poisson structure on the moduli space, due to Tyurin and Bottacin, is taken as external input. The main result (Theorem 5.1) states that, on the open subset M'_0 parametrizing sheaves E = i_*L with smooth support C meeting D in finitely many points, the fibers of the natural morphism phi0: M'_0 -> Sym^{d'}(D), E -> C∩D, are the symplectic leaves of the Poisson structure. The proof identifies, at each point, the tangent space to the fiber of phi0 with the symplectic subspace T0 of Ext^1(E,E) determined by the factorization of the Poisson map sigma, using the local-to-global spectral sequence and the identification of the tangent space of the Hilbert scheme with H^0(N_{C/S}). A vanishing statement H^0(C,K_S|_C)=0 is used to prove smoothness of the Hilbert scheme at the relevant points.","tokens_in":7440,"tokens_out":5643,"duration_ms":56598,"significance":"If the result is correct, it gives a clean geometric description of the symplectic foliation of the Tyurin–Bottacin Poisson structure on moduli of pure dimension-one sheaves: leaves are controlled by the intersection of the Fitting subscheme F(E) with the anticanonical divisor D. The paper is concise, relies transparently on the external Poisson-structure results of Tyurin and Bottacin, and contains a concrete tangent-space computation that is the core of the argument. The main conceptual contribution—relating the symplectic subspace to deformations fixing the intersection C∩D—is plausible and useful. The missing connectivity and smoothness arguments are gaps in the proof of the stated theorem, but they appear fixable without changing the main computation.","major_comments":[{"comment":"The proof establishes that, for each point E, the tangent space to the fiber of phi0 equals the symplectic subspace T0. This shows the fiber is an integral submanifold of the symplectic distribution, but a symplectic leaf is required to be a maximal connected integral submanifold. The connectedness of the fibers of phi0 is never proved. In particular, the fibers of g: Q0 -> Sym^{d'}(D) could in principle be disconnected, and if so a fiber of phi0 would split into several symplectic leaves, making the statement of Theorem 5.1 false as written. The manuscript should add a proof that g^{-1}(Z) and hence phi0^{-1}(Z) is connected, for instance by realizing g^{-1}(Z) as a connected linear system (a projective space or a connected open subset of one) and using that the fibers of f0 are Picard varieties, which are connected.","section":"Section 5, Theorem 5.1"},{"comment":"The assertion that f0 is a smooth morphism is not proved. To use the short exact sequence (5.1) for tangent spaces, one needs f0 to be smooth, or at least to know that the differential of f0 is surjective and that the relative tangent sequence is exact. The identification of the differential df0|E with the map i_{E,E} in the local-to-global sequence is a plausible route to pointwise surjectivity, but the argument should be written out explicitly. Similarly, the identification of the tangent space to the fiber of g with H^0(N_{C/S}(-D)) needs justification: one must explain why the infinitesimal deformations of C fixing the subscheme Z = C∩D are exactly these sections, and that g is a well-behaved morphism on the relevant open subset.","section":"Section 5, before (5.1)"},{"comment":"The statement that H^0(C,K_S|_C)=0 for every effective curve C with C·D≠0 is false when C is reducible and has an irreducible component disjoint from D: on that component, K_S is trivial, so the restriction has nonzero global sections. The vanishing is correct for the smooth curves appearing in Q0, since for a smooth connected C with C·D>0 the line bundle O_C(-D) has negative degree. The overbroad wording should be corrected to apply only to the smooth curves used in the proof, although this error does not invalidate Theorem 5.1 itself.","section":"Section 5, smoothness of the Hilbert scheme"}],"minor_comments":[{"comment":"The abstract states that the symplectic leaves of the full moduli space M_H(S,P) are the fibers of the natural map to the symmetric power of D, but the theorem is proved only for the open subset M'_0 where F(E) is smooth and intersects D in a zero-dimensional subscheme. The abstract should be qualified to match the theorem, since the map is not defined on the whole moduli space.","section":"Abstract and Introduction"},{"comment":"The letter C is used both for a divisor class in NS(S) (Section 2 and later in M = M_H(S,C,P)) and for a specific curve F(E) (Section 5). Also d is used for the degree of the line bundle L while d' = C·D. This is confusing; please introduce distinct notation, for example [C] for the class and C for the curve.","section":"Notation throughout"},{"comment":"In the proof of Lemma 3.1, the claim that K_S|_C is trivial when C·D=0 should be justified explicitly: since s vanishes exactly on D and C∩D is empty (because the intersection is zero-dimensional and has degree zero), the restriction of s to C is nowhere vanishing, so K_S|_C is trivial. As written, 'it follows' skips the key point.","section":"Lemma 3.1"},{"comment":"The diagram (4.1) is difficult to read because several rows and columns are aligned ambiguously; for example the inclusion on the right column is not clearly labeled. Please redraw the diagram with explicit labels for all arrows, especially the vertical map sigma_2 and the inclusion H^0(Ext^1(E,E⊗K_S)) -> H^0(Ext^1(E,E)).","section":"Proposition 4.2, diagram (4.1)"},{"comment":"The definition of the open subset M'_0 says 'parametrizing sheaves E with F(E) smooth'; this should say that F(E) is a smooth curve (and that F(E)∩D is zero-dimensional, as already stated). It may also be worth noting that smoothness of the Fitting subscheme is an open condition, so this is indeed a Zariski open subset.","section":"Section 2, definition of M'_0"}],"recommendation":"major_revision","confidential_remarks":"The paper's advertised result is stronger than what is proved: Theorem 5.1 requires connectedness of fibers to conclude they are symplectic leaves, and this is not addressed. The tangent-space computation is sound and the gaps are likely fixable, so I do not recommend rejection, but the revision needs to supply the missing connectivity and smoothness arguments. Please also ensure the abstract is aligned with the actual scope of the theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main result is very likely true, and the proof's core computation is right. The paper identifies the symplectic leaves of the Tyurin–Bottacin Poisson structure on moduli of pure dimension one sheaves over a Poisson surface: they are fibers of the map sending a sheaf to the intersection of its Fitting support with the degeneracy divisor D. That is new for general Poisson surfaces (Mukai handled the symplectic case; Tyurin and Bottacin constructed the Poisson structure but did not describe the leaves). The method is explicitly modeled on Markman and Bottacin's work for Higgs pairs, and the local tangent-space identification is elegant: the symplectic subspace T0 in Ext^1(E,E) is exactly the tangent space to the fiber of φ0, via the local-to-global spectral sequence and the identification of deformations of C fixing C∩D with H^0(N_{C/S}(-D)). This is the heart of the paper and it is sound.\n\nWhere the paper is soft is the step from tangent-space equality to \"fibers are symplectic leaves.\" The proof shows, at each point, that T E φ0 = T0. That makes each connected component of a fiber an integral submanifold of the symplectic distribution, but a symplectic leaf is a maximal connected submanifold. If a fiber of φ0 is disconnected, it splits into several leaves, and the statement as written is false. The paper never proves that the fibers of g: Q0 → Sym^{d'}(D) are connected, nor that the relative Picard fibers over them are connected. This is not a fatal flaw — it is likely fixable by showing g^{-1}(Z) is connected — but it is load-bearing for the final formulation.\n\nThree smaller issues. The claim in Section 5 that H^0(C,K_S|_C)=0 for every effective curve with C·D ≠ 0 is false for reducible curves with a component disjoint from D. Only smooth curves are needed for the open subset under consideration, so this is a wording error, not a mathematical one. The smoothness of f0 is asserted rather than proved; smooth total space plus smooth fibers does not by itself imply smoothness of the morphism, though standard deformation theory should give it. And the notation sometimes conflates the curve class C with a representative curve, which makes the paper harder to read.\n\nWho is this for? People working on Poisson moduli spaces or on the analogy between Higgs bundles and sheaves on surfaces. It is a short, credible structural result that deserves serious refereeing. I would send it to a referee, asking for a proof or disproof of connectedness of the fibers and a proper justification that f0 is smooth; those are likely minor revisions, not grounds for rejection.","headline":"A probably correct and clean leaf computation for Tyurin–Bottacin Poisson moduli, but the theorem as stated needs a connectedness argument before it is fully proved.","tokens_in":7998,"tokens_out":5280,"would_cite":true,"duration_ms":55980,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14J60","53D17","32J15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The symplectic leaves of the Poisson moduli space of pure dimension-one sheaves are exactly the fibers of the map sending each sheaf to the intersection of its Fitting support with the degeneracy divisor.","keywords":["Poisson surface","torsion sheaf","moduli space of sheaves","symplectic leaf","pure dimension one sheaf","Fitting ideal","Poisson moduli space","Hilbert polynomial"],"falsifier":"Exhibit a fiber of $g:Q_0\\to\\operatorname{Sym}^{d'}(D)$ with more than one connected component; then the corresponding fiber of $\\phi_0$ is disconnected and cannot be a single symplectic leaf under the standard definition, contradicting Theorem 5.1 as stated.","tokens_in":6945,"feed_emoji":"🍃","tokens_out":11318,"duration_ms":109299,"temperature":0.7,"pith_summary":"The paper studies the moduli space $M_H(S,P)$ of stable sheaves of pure dimension one on a smooth complex projective surface $S$ carrying a Poisson structure coming from a section $s$ of the anticanonical bundle. Its main claim is that the symplectic leaves of this moduli space are precisely the fibers of a natural morphism: send a sheaf $E$ to the zero-dimensional intersection $F(E)\\cap D$, where $F(E)$ is the Fitting support of $E$ and $D$ is the divisor where $s$ vanishes. If this is true, the entire symplectic foliation of the moduli space is captured by one discrete geometric datum, the position of the support curve relative to $D$, and the symplectic form along a leaf is nondegenerate exactly on deformations that leave that intersection fixed. This matters because it turns an abstract Poisson geometry question into a concrete statement about curves on a surface and their intersections with a fixed divisor.","feed_headline":"Symplectic leaves equal fibers that fix support–divisor intersections","feed_subtitle":"On a Poisson surface, each maximal symplectic submanifold is determined by where a sheaf's support meets the degeneracy divisor.","key_machinery":"The central mechanism is the factorization of the Poisson map into a surjection onto a symplectic subspace $T_0$. For a Poisson structure $\\sigma:T^*\\to T$, $T_0=\\sigma(T^*)$ carries the induced symplectic form, and the paper identifies this $T_0$ with the tangent space to the fiber of $\\phi_0$. That identification is made through the local-to-global spectral sequence for Ext: the bottom row of the paper's diagram (4.1), $H^1(\\mathcal{H}om(E,E))\\to\\operatorname{Ext}^1(E,E)\\to H^0(\\mathcal{E}xt^1(E,E))$, is matched with the tangent sequence of the map $f_0$ sending $E$ to its Fitting support in the Hilbert scheme, while the top-right term $H^0(N_{C/S}(-D))$ is matched with the tangent space to the fibers of the intersection map $g$. The Fitting support $F(E)$ is the subscheme cut out by the zeroth Fitting ideal, and for $E=i_*L$ with smooth $C$ it is simply $C$.","core_discovery":"For sheaves $E=i_*L$ whose Fitting support $F(E)$ is a smooth curve $C$ meeting $D$ in finitely many points, define $\\phi_0(E)=C\\cap D$ as a point of the symmetric product $\\operatorname{Sym}^{d'}(D)$, where $d'=C\\cdot D$. The paper's theorem states that the fibers of $\\phi_0$ are the symplectic leaves of the Poisson structure on the open subset $M'_0$. The proof shows that at each such $E$, the tangent space to the fiber equals the symplectic subspace $T_0$ of $\\operatorname{Ext}^1(E,E)$ obtained by factoring the Poisson map $\\sigma:\\operatorname{Ext}^1(E,E\\otimes K_S)\\to\\operatorname{Ext}^1(E,E)$. In particular, the leaf direction is characterized infinitesimally by preserving both the support curve and its intersection with the degeneracy divisor.","pith_inferences":["If the tangent identification is uniform over $M'_0$, the Poisson structure on that open locus should be regular, with all symplectic leaves of the same dimension rather than a foliation whose leaf dimension jumps.","The statement may need refinement when a fiber of $g:C\\mapsto C\\cap D$ is disconnected: under the usual definition of a symplectic leaf as a maximal connected submanifold, each connected component would be a leaf, and the whole fiber would be a union of leaves rather than one leaf.","A similar argument could plausibly extend to sheaves whose Fitting support is singular or has nilpotent thickenings, but the tangent identification would have to be redone because the Hilbert scheme of such supports need not be smooth."],"forward_implications":["For every point $E=i_*L$ with smooth support $C$ and $C\\cap D$ finite, the symplectic leaf through $E$ lies inside the fiber of $\\phi_0$, consisting of sheaves whose support meets $D$ in the same zero-dimensional subscheme.","Infinitesimally, a deformation of $E$ lies in the symplectic leaf exactly when it preserves the subscheme $C\\cap D$, which is the geometric content of the equality of $T E\\phi_0$ with the symplectic subspace $T_0$.","When $C\\cdot D=0$, the intersection datum is empty and the corresponding open moduli space is symplectic: the Poisson map is an isomorphism, so the whole open component is a single symplectic leaf.","In the spectral-cover description of Higgs bundles, the map $E\\mapsto F(E)$ is the analogue of the Hitchin map, so the theorem describes the symplectic foliation in that setting by the relative position of the spectral curve and the divisor $D$."],"supporting_citations":[{"why":"Constructs the skew-symmetric morphism $\\sigma$ on the moduli space of sheaves over a Poisson surface, the Poisson structure studied here.","marker":"[Ty]"},{"why":"Proves that the Schouten–Nijenhuis bracket of $\\sigma$ vanishes, so the construction indeed yields a Poisson structure.","marker":"[Bo2]"},{"why":"Treats the abelian or K3 case where the Poisson map is an isomorphism, giving the base symplectic case that the paper compares with.","marker":"[Mu]"},{"why":"Provides the analogous symplectic-leaf statement for moduli of stable pairs, which the paper's Theorem 5.1 is modelled on.","marker":"[Bo1]"},{"why":"Gives the spectral-curve formulation and the analogous result for Higgs pairs, used for the Hitchin-map analogy and the tangent identifications.","marker":"[Ma]"},{"why":"Provides the moduli space of stable sheaves on which the whole construction takes place.","marker":"[Si]"},{"why":"Provides the compactified Picard scheme used to identify the fibers of $f_0$ with Picard varieties of line bundles on the support curve.","marker":"[AK]"}],"fun_headline_variants":["Symplectic leaves are fibers of support–divisor intersection map","Leaves of Poisson moduli equal fibers that fix support-divisor contact","Symplectic leaves determined by sheaf support's divisor intersections","Poisson moduli: symplectic leaves are fibers of support–divisor map"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes that the two maps defining the fiber—the map from sheaves to their support curves and the map from support curves to their intersections with $D$—are smooth enough for tangent-space sequences to describe the fibers, and that each resulting fiber is connected so it is a single maximal symplectic leaf rather than several.","fun_headline_variants_meta":{"raw":{"variants":["Symplectic leaves are fibers of support–divisor intersection map","Leaves of Poisson moduli equal fibers that fix support-divisor contact","Symplectic leaves determined by sheaf support's divisor intersections","Poisson moduli: symplectic leaves are fibers of support–divisor map"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000502,"raw_usage":{"total_tokens":2392,"prompt_tokens":824,"completion_tokens":1568,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":440,"completion_tokens_details":{"reasoning_tokens":1489}},"tokens_in":440,"tokens_out":1568,"duration_ms":11764,"temperature":1.0,"reasoning_tokens":1489,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:45:08.918292+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a fiber of $g:Q_0\\to\\operatorname{Sym}^{d'}(D)$ with more than one connected component; then the corresponding fiber of $\\phi_0$ is disconnected and cannot be a single symplectic leaf under the standard definition, contradicting Theorem 5.1 as stated.","supporting_citations":[],"review_version":1}