{"id":"24cc6639-9b40-484b-b9a8-cc99e8ee96ab","arxiv_id":"2607.06189","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"An explicit Q-algebra automorphism of symmetric functions, given by q-deformations of power sums, converts the S_n-equivariant Euler characteristics of stable map moduli spaces to those of ε-stable quasimap moduli spaces over Grassmannians.","lead":"The paper proves an explicit algebraic formula relating the Euler characteristics of moduli spaces of stable maps and ε-stable quasimaps to Grassmannians, as the stability parameter ε varies. A generalist might read it to see how sophisticated combinatorial machinery tames the geometry of curve-counting spaces that are otherwise too singular to study directly.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"No significant objection identified. The Artin stack extension in Lemma 4.19 is handled by standard smooth-chart techniques, and the fiber diagram is a genuine pullback because the quasimap stability condition on contracted rational bridges matches the weighted-stable-map condition exactly.","rationale":"The reader correctly identifies Theorem 4.14 / Lemma 4.19 as the key geometric input and the place where the argument is least detailed. However, upon careful examination, the argument is sound. The extension to Artin stacks uses the standard technique of passing to smooth charts, which is well-established. More importantly, the fiber diagram in Lemma 4.19 is a genuine pullback square because the quasimap stability condition on contracted rational bridges (Definition 2.1(4)) is equivalent to the weighted-stable-map condition (Definition 2.5(4)) when the support of the torsion sheaf Q is identified with the ε-weighted markings. Specifically, for a contracted rational bridge T, the quasimap condition gives ε·deg(Q|_T) > 0, which is exactly the requirement that at least one ε-weighted point lies on T. Since the right vertical arrow is an iterated vector bundle (Proposition 4.7), the left vertical arrow (a base change) inherits this structure. The algebraic steps in Corollary 4.25 are verified: the substitution p^{(k)}_j → (L^{k-2}q)^j correctly encodes both the L-factors from the Quot scheme formula and the degree tracking, and the final expression p_n + Σ q^{nk}[r+k-1 choose k]_L^n follows from standard properties of Adams operations and the Serre characteristic of Sym^k(P^{r-1}). The plethystic formalism (§3, §5) and genus-zero input (§5.3, from [Bag22, Str81]) rest on established results. The paper would benefit from explicitly stating that the fiber diagram is a pullback and explaining the matching of stability conditions, but this is an exposition issue rather than a correctness gap. The CONDITIONAL verdict is slightly conservative; the argument is more secure than the reader suggests. However, since no formal verification or independent computational check is provided, maintaining CONDITIONAL is defensible.","tokens_in":33087,"tokens_out":14378,"duration_ms":980210,"concrete_test":"Independently compute χ(Q^{1/2}_{1,0}(G(2,N),1)) via C*-localization on the quasimap space (following the fixed-point analysis in [Tod11, §5.1] and [MOP11, §7.3], which involves graphs decorated by torsion sheaf data) and compare with the value 4·C(N,2) predicted by Table 2 via Theorem A. Agreement would confirm the key geometric input (Theorem 4.14) in a case where the Quot scheme stratification over the Artin stack is nontrivial.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's concern centers on Theorem 4.14 and Lemma 4.19: whether the relative Quot scheme formula (Proposition 4.7, proved for families of smooth curves over a scheme base) extends correctly to the Artin stack of prestable maps, and whether the stability condition (Definition 4.16) is compatible with the iterated vector bundle stratification. Upon careful examination, the argument holds. (1) Extension to Artin stacks: Lemma 4.19 observes that Lemma 4.4 (bundle independence) extends to Artin stacks by passing to smooth charts. This is standard: an Artin stack admits a smooth surjective cover by a scheme, and the Quot scheme, being of finite presentation, descends. The stratifications trivializing the bundle E exist locally on charts and glue. (2) The fiber diagram in Lemma 4.19 is a genuine pullback square. The support map Quot^{k,ε}(E) → Conf^k_ε/S_k sends a quotient to its support divisor. The stability condition (Definition 4.16) requires that Q has positive degree on every contracted rational bridge. For a rational bridge T (g=0, n_T=2) contracted by f (deg(f|_T)=0), the quasimap stability condition (Definition 2.1(4)) gives 2(0)-2+2+ε·deg(V*|_T) > 0, i.e., ε·deg(Q|_T) > 0, i.e., deg(Q|_T) > 0. This is exactly the condition that the support of Q meets T, which is the weighted-stable-map condition requiring at least one ε-weighted marking on every rational bridge. Thus the stable locus Quot^{k,ε,stab} is precisely the preimage of M^{nrt}_{g,n|εk}/S_k under the support map, making the square a pullback. (3) Since the right vertical arrow (from Proposition 4.7) is an iterated vector bundle, the left vertical arrow (a base change) is also an iterated vector bundle. The L-factors and degree tracking are correctly encoded in the substitution of Corollary 4.23, as verified by tracing through the Adams operations and Corollary 3.20. The plethystic inversion (Lemma 5.4) and rational tails separation (Lemma 5.3) rest on established techniques from [GP06, KS26a]. No gap","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper proves a wall-crossing formula (Theorem A) relating the S_n-equivariant Euler characteristics, as virtual mixed Hodge structures, of moduli spaces of stable maps and ε-stable quasimaps to Grassmannians G(r,N). The formula is encoded by an explicit Q-algebra automorphism B_{r,ε} of the ring of symmetric functions, defined via q-deformations of power sum symmetric functions using q-binomial coefficients evaluated at L = [H²(P¹;Q)]. The proof follows a three-step strategy: (1) plethystic reduction to no-rational-tails loci (Lemma 5.3), (2) stratification of quasimap spaces via relative Quot schemes over the Artin stack of prestable maps (Theorem 4.14), and (3) symmetric function wall-crossing for weighted stable maps (Corollary 3.16). The ε→0⁺ limit recovers the Marian–Oprea–Pandharipande stable quotient moduli spaces. The paper also provides explicit genus-one calculations via torus localization and walks in the Johnson graph.","tokens_in":33380,"tokens_out":3086,"duration_ms":197145,"significance":"The result is new even for r=1 (target P^{N-1}) and provides the first motivic/topological incarnation of quasimap wall-crossing formulas for Grassmannians. The automorphism B_{r,ε} is parameter-free: it is defined by q-binomial coefficients evaluated at L = [H²(P¹;Q)], the Serre characteristic of P¹, not by fitted constants. The genus-zero input M_{0,r,N} comes from Bagnarol's independent computation [Bag22] via Strømme's cell decomposition, so the formula is both invertible and computationally effective. The relative Quot scheme formula (Proposition 4.7) for families of smooth curves may be of independent interest. The genus-one calculations in Section 6 yield explicit closed-form Euler characteristics (Tables 1–2) and are falsifiable. The paper ships concrete, checkable predictions.","major_comments":[{"comment":"Lemma 4.19, proof: The extension of the bundle-independence result (Lemma 4.4) from families of smooth curves over schemes to the Artin stack of prestable maps M_nrt_{g,n}(G(r,N),d) is the key geometric step underpinning Theorem 4.14 and hence Theorem A. The proof states that 'by passing to smooth charts of C and S if necessary, there still exist finite locally closed stratifications' satisfying the same requirements as in Lemma 4.4, and that the fibers over strata are isomorphic. While the smooth-chart descent argument is standard for the Quot scheme itself, the paper does not explicitly verify that the stratifications trivializing the bundle E are compatible with the stability condition of Definition 4.16 across strata. The fiber diagram in the proof is asserted to imply that fibers of Quot^{k,ε,stab}(ev*S) and Quot^{k,ε,stab}(O^⊕r) over strata of M_nrt_{g,n|εk}/S_k are isomorphic, but","section":null},{"comment":"Proposition 4.7: The statement involves both a family of prestable curves C→S with sections s₁,...,sₙ and a family of smooth curves C̄→S 'obtained by deleting the nodes and any subset of the sections from C,' with the formula stated for Quot^{k,ε}_{C̄/S}(O^r_{C̄}). The role of C and its sections in the statement is unclear, since the formula only concerns C̄. If the intent is to set up notation for later applications (where C is the universal prestable curve and C̄ its smooth locus), this should be clarified; otherwise the prestable curve C appears to play no role in the proposition itself.","section":null}],"minor_comments":[{"comment":"§4.2, Definition 4.15: The notation M_nrt_{g,n}(G(r,N),d) for the Artin stack of prestable maps without rational tails is potentially confusing, as the superscript 'nrt' was used in Definition 2.4 for a locally closed substack of the moduli of ε-stable quasimaps. A brief remark distinguishing the two uses would help.","section":null},{"comment":"§4, proof of Proposition 4.7, footnote 2: The authors acknowledge an abuse of notation where the strata Quot^{ℓ₁,...,ℓᵣ} coincide with those from Lemma 4.4 'a priori not' being the same. This should be reconciled or the notation differentiated.","section":null},{"comment":"§5.3, Theorem 5.10: The formula for M_{0,r,N} involves (Quot_{0,P¹})⁻¹. It would help the reader to note explicitly that this inverse exists in K⁰(MHS)⊗Λ[[q]] (i.e., that Quot_{0,P¹} is a unit in the relevant ring), or to clarify the sense in which the inverse is taken.","section":null},{"comment":"Tables 1–2: The entries use the notation (N choose k) without specifying that this is the ordinary binomial coefficient (as opposed to the q-binomial). A brief note in the table caption would prevent confusion.","section":null},{"comment":"§6.2, Proposition 6.3: The notation A_g, Ȧ₀, A″₀ is introduced but the definitions of Ȧ₀ and A″₀ (derivatives/specializations of A_g) are not stated explicitly in the manuscript; they appear to reference [KS26a]. A brief definition or pointer would make the proposition self-contained.","section":null},{"comment":"The reference [KS26a] is listed as 'to appear' in Crelle's Journal; if it has appeared by the time of revision, the reference should be updated.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The reader's report and stress-test note both flag Lemma 4.19 as the potential weak point. On careful reading, the argument is correct: the stability condition (Definition 4.16) on contracted rational bridges matches the quasimap stability condition (Definition 2.1(4)) exactly, so the fiber diagram is a genuine pullback and the fiber-by-fiber isomorphisms do patch. The concern is one of exposition rather than correctness. I recommend minor revision with a request to expand the proof of Lemma 4.19. The paper is a strong contribution and well within the scope of a serious algebraic geometry journal."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and for identifying two points where the exposition can be improved. Both comments are well-taken and will be addressed in the revision. The first concerns a gap in the verification that stratifications trivializing the bundle are compatible with the stability condition in Definition 4.16; we will add the missing argument. The second concerns a notational redundancy in the statement of Proposition 4.7; we will clarify the role of the prestable curve C.","responses":[{"response":"The referee is correct that the proof of Lemma 4.19 as written does not explicitly verify the compatibility between the stratifications trivializing the bundle E and the stability condition of Definition 4.16. We will add the missing argument in the revision. The key observation is as follows. The stability condition in Definition 4.16 is an open condition on the relative Quot scheme over the Artin stack M^nrt_{g,n}(G(r,N),d-k): it requires that if a rational bridge T is contracted by the map f, then the quotient sheaf Q has positive degree along T. This condition depends only on the underlying prestable map [f:C→G(r,N)] and the support of the torsion sheaf Q, not on the choice of trivialization of the tautological bundle ev*S. More precisely, the isomorphisms (4.5) in the proof of Lemma 4.4 are constructed via extension by zero from open sets on which E is trivial, and these isomorphisms preserve the support of the quotient sheaf. Since the stability condition of Definition 4.16 is defined in terms of the degree of Q along contracted rational bridges—which is determined by the support of Q—the isomorphisms between fibers of Quot^{k,ε}(ev*S) and Quot^{k,ε}(O^⊕r) over each stratum automatically restrict to isomorphisms between the stable loci. We will spell out this argument explicitly in the revised proof.","revision_made":"yes","referee_comment":"Lemma 4.19, proof: The extension of the bundle-independence result (Lemma 4.4) from families of smooth curves over schemes to the Artin stack of prestable maps is the key geometric step. The proof states that 'by passing to smooth charts of C and S if necessary, there still exist finite locally closed stratifications' satisfying the same requirements, but does not explicitly verify that the stratifications trivializing the bundle E are compatible with the stability condition of Definition 4.16 across strata. The fiber diagram is asserted to imply that fibers of Quot^{k,ε,stab}(ev*S) and Quot^{k,ε,stab}(O^⊕r) over strata are isomorphic."},{"response":"The referee is correct that the role of the prestable curve C and its sections is not clear from the statement of Proposition 4.7 as written. The intent is indeed to set up notation for later applications: in Section 4.2, C is the universal prestable curve over the Artin stack of prestable maps, and C̄ is its smooth locus (the complement of markings and nodes). The formula itself only concerns C̄, but we state it in the more general context of a prestable curve C with sections to make the later specialization to the universal curve transparent. We will revise the statement to clarify this: we will restructure the proposition so that C̄ is the primary object (a flat family of smooth curves over S), and add a remark explaining that in applications, C̄ arises as the smooth locus of a universal prestable curve C over the Artin stack M^nrt_{g,n}(G(r,N),d).","revision_made":"yes","referee_comment":"Proposition 4.7: The statement involves both a family of prestable curves C→S with sections s₁,...,sₙ and a family of smooth curves C̄→S 'obtained by deleting the nodes and any subset of the sections from C,' with the formula stated for Quot^{k,ε}_{C̄/S}(O^r_{C̄}). The role of C and its sections in the statement is unclear, since the formula only concerns C̄. If the intent is to set up notation for later applications, this should be clarified."}],"tokens_in":32926,"tokens_out":908,"duration_ms":169408,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"This paper proves the first motivic-level wall-crossing formula between stable maps and ε-stable quasimaps to Grassmannians. The main result (Theorem A) gives an explicit, invertible automorphism B_{r,ε} on symmetric functions — defined by q-binomial coefficients evaluated at L = [H²(P¹;Q)] — that converts the S_n-equivariant Serre characteristic generating function for stable maps into the one for ε-stable quasimaps. The ε→0 limit recovers stable quotients. This is a genuine extension of the enumerative wall-crossing program (CFK, Zhou) to mixed Hodge structures, and it's new even for P^{N-1} (r=1). The operators are computable, the genus-one Euler characteristics in Tables 1–2 are new, and the derivation is parameter-free throughout. The three-step proof strategy is clean: plethystic reduction to no-rational-tails loci, stratification via relative Quot schemes, then symmetric function wall-crossing for weighted stable maps. Steps 1 and 3 rest on established techniques (Getzler–Pandharipande, Bagnarol, prior work of the authors). The relative Quot scheme formula (Proposition 4.7) is a nice extension of Bifet–Bagnarol–Fantechi–Perroni–Ricolfi to the relative and ε-weighted setting, proved by an iterated vector bundle stratification in the style of Mochizuki. The one soft spot is the extension from smooth curves over a scheme to the Artin stack of prestable maps (Theorem 4.14, Lemma 4.19). The reader flagged this as a conditional concern. On careful reading, I think the argument actually holds. The bundle-independence lemma (Lemma 4.4) extends to Artin stacks by passing to smooth charts — this is standard. More importantly, the fiber diagram in Lemma 4.19 is a genuine pullback: the quasimap stability condition on a contracted rational bridge T gives ε·deg(Q|_T) > 0, which is exactly the condition that the support divisor meets T, matching the weighted-stable-map condition. So the stable locus is precisely the preimage of the weighted-stable-map space under the support map, and base change preserves the iterated vector bundle structure. The concern is really about exposition — the authors sketch this reduction rather than spelling it out — not about a mathematical gap. I'd ask for a more detailed writeup of this step, but it doesn't undermine the result. This paper is for algebraic geometers working on motivic invariants of moduli spaces, wall-crossing, or quasimap theory. It deserves a serious referee. The result is correct and new, and the main thing to check carefully in review is the Artin stack extension, which I believe will hold up.","headline":"Motivic quasimap wall-crossing for Grassmannians — solid paper, one exposition gap worth flagging","tokens_in":34102,"tokens_out":1275,"would_cite":true,"duration_ms":89371,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Explicit automorphism relates Euler characteristics of stable maps and quasimaps","keywords":[],"falsifier":"If the fiber diagrams in the proof of Lemma 4.19 fail to commute for certain strata of the prestable map stack, the equality of Grothendieck ring classes in Theorem 4.14 would not hold, and the operators B_{r,epsilon} would not correctly relate the no-rational-tails generating functions, breaking the main formula.","tokens_in":33270,"feed_emoji":"🔄","tokens_out":1436,"duration_ms":232482,"temperature":0.7,"pith_summary":"The paper proves that for any genus g at least 1 and any rational parameter epsilon at least 0, the generating functions for the S_n-equivariant Euler characteristics of moduli spaces of stable maps to a Grassmannian G(r,N) and of epsilon-stable quasimaps to the same target are related by an explicit, invertible algebra automorphism B_{r,epsilon} of the ring of symmetric functions tensored with the Grothendieck group of mixed Hodge structures. This automorphism acts by q-deforming the power sum symmetric functions, sending p_j to p_j plus a finite (or infinite, when epsilon goes to 0) sum whose coefficients are L-binomial coefficients built from the class L of the affine line and the geometry of projective space. The formula is stated as Theorem A: after composing each generating function with a genus-zero correction term involving plethysm by the derivative of the genus-zero generating function divided by the Euler characteristic of the Grassmannian, the automorphism B_{r,epsilon} carries the stable-map side to the quasimap side. When epsilon tends to 0, the formula specializes to a relation between stable maps and the Marian-Oprea-Pandharipande moduli space of stable quotients. The proof proceeds in three stages: first, a plethystic argument separates contributions from curves with no rational tails; second, the key geometric input stratifies the quasimap moduli space by relative Quot schemes of zero-dimensional quotients over the Artin stack of prestable maps, and computes their motives in terms of moduli spaces of weighted stable maps; third, symmetric function theory handles wall-crossing among weighted stable map spaces, producing the operators B_{r,epsilon}. The paper also works out the genus-zero case explicitly, making the full transformation computable and invertible, and provides closed-form graph-enumeration formulas for the topological Euler characteristic in genus one.","feed_headline":"Explicit automorphism relates Euler characteristics of stable maps and quasimaps","feed_subtitle":"An invertible q-deformation of symmetric functions bridges three families of moduli spaces over Grassmannians, with closed-form genus-one计算","key_machinery":"The proof combines three mechanisms: (1) plethystic decomposition of moduli spaces into no-rational-tails loci and genus-zero tail contributions, using the Frobenius characteristic and S-space formalism of Getzler-Pandharipande; (2) a stratification of epsilon-stable quasimap moduli spaces by relative Quot schemes of zero-dimensional quotient sheaves over the Artin stack of prestable maps, whose motives are computed via iterated vector bundle structures over weighted configuration spaces (Proposition 4.7 and Theorem 4.14); (3) wall-crossing formulas for weighted stable maps encoded by fundamental transformations of symmetric functions (Corollary 3.16), which assemble into the operators B_{r,","core_discovery":"The central discovery is that the motivic and topological Euler characteristics of three families of moduli spaces—stable maps, epsilon-stable quasimaps, and stable quotients to Grassmannians—are all governed by a single explicit automorphism of the ring of symmetric functions. The automorphism B_{r,epsilon} sends each power sum generator p_j to p_j plus a sum of L-binomial coefficients multiplied by powers of q, where L is the Hodge-theoretic class of the affine line. This operator encodes the geometric operation of replacing marked points (carrying symmetric group actions) with punctual Quot schemes supported at those points, with the parameter epsilon controlling how many points may_COLL.","pith_inferences":["The operators B_{r,epsilon} depend on the target only through the rank r and the class L, suggesting that analogous wall-crossing automorphisms might exist for other GIT targets whose motivic zeta functions have similar product structures, such as flag varieties or certain spherical varieties.","The appearance of L-binomial coefficients e(Sym^k(P^{r-1})) in B_{r,epsilon} connects the wall-crossing formula to the motivic zeta function of P^{r-1}, raising the question of whether deeper arithmetic properties of these zeta functions (such as functional equations or special values) translate into symmetries among the Euler characteristics of the moduli spaces.","The graph-enumeration approach in genus one, which reduces to counting walks in the Johnson graph, could in principle extend to higher genus via more elaborate decorated graph sums, though the combinatorics of torsion sheaves at fixed points would introduce additional complexity not present in the stable map case."],"forward_implications":["The automorphism B_{r,epsilon} provides a computable and invertible bridge: knowing the Euler characteristic of the stable map space for all n and d determines that of every epsilon-stable quasimap space, and vice versa, without computing the quasimap spaces directly.","In genus one, the paper combines the wall-crossing formula with torus localization and walk-counting on the Johnson graph J(r,N) to produce closed-form expressions for the topological Euler characteristic of quasimap spaces Q^epsilon_{1,n}(G(r,N),d) for arbitrary epsilon, n, r, N, and d.","The epsilon to 0 limit gives a new motivic wall-crossing formula between stable maps and the Marian-Oprea-Pandharipande stable quotient spaces, specializing the general quasimap story to a previously studied moduli problem.","For r=1 (target projective space), the existence of contraction morphisms yields an alternative, simpler wall-crossing formula (Corollary 5.16) that avoids the L-factors appearing in the general Grassmannian case."],"fun_headline_variants":["One automorphism links Euler characteristics of maps, quasimaps, and quotients","Symmetric function automorphism encodes moduli Euler characteristics","Wall-crossing for Grassmannian moduli via q-deformed symmetric functions","Explicit automorphism governs motivic Euler characteristics for Grassmannians","q-deformed symmetric functions bridge stable maps and quasimaps"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The key geometric input, Theorem 4.14, extends a formula for the motive of relative Quot schemes from the case of smooth curves to the case of prestable maps over an Artin stack, by asserting that the stratification by iterated vector bundles and the stability condition interact correctly so that fiber-by-fiber isomorphisms across strata commute as claimed. This extension is sketched by analogy with the smooth-curve case but the verification across strata of the Artin stack, ","fun_headline_variants_meta":{"raw":{"variants":["One automorphism links Euler characteristics of maps, quasimaps, and quotients","Symmetric function automorphism encodes moduli Euler characteristics","Wall-crossing for Grassmannian moduli via q-deformed symmetric functions","Explicit automorphism governs motivic Euler characteristics for Grassmannians","q-deformed symmetric functions bridge stable maps and quasimaps","Wall-crossing formula connects stable maps, quasimaps, and quotients"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1321,"prompt_tokens":600,"completion_tokens":721,"prompt_tokens_details":null},"tokens_in":600,"tokens_out":721,"duration_ms":29378,"temperature":1.0,"reasoning_tokens":630,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T13:47:37.364798+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the fiber diagrams in the proof of Lemma 4.19 fail to commute for certain strata of the prestable map stack, the equality of Grothendieck ring classes in Theorem 4.14 would not hold, and the operators B_{r,epsilon} would not correctly relate the no-rational-tails generating functions, breaking the main formula.","supporting_citations":[],"review_version":1}