{"id":"7e9a2983-140e-4568-945c-ef0aa0939077","arxiv_id":"2411.16141","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new extended weighted blow-up construction yields proper Deligne-Mumford compactifications of stable maps to quotient stacks with projective good moduli space.","lead":"This paper proves that moduli spaces of maps from curves into many quotient stacks admit proper Deligne-Mumford compactifications, by enlarging the target stack with a new birational operation. The construction is applied to fibered log Calabi-Yau pairs, several GIT quotients, and a conjecture of Hassett.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1 depends entirely on the unstated [ER21] saturated-blow-up theorem; if that theorem's hypotheses differ from 'dense properly stable locus', the enlargement construction is unsupported.","rationale":"The reader's weakest-assumption identification points to the same spot: Theorem 5.1 is the novel enlargement step, and it is bootstrapped from [ER21] without proving anything about that sequence. I read the full manuscript in good faith. The extended weighted blow-up formalism, the induction in Theorem 5.1, the relative GIT Proposition 2.16, and the later applications in Section 6 and Appendix A are coherent conditional on [ER21] delivering a finite sequence of saturated blow-ups ending in a Deligne-Mumford stack. I found no internal contradiction. The concern is therefore not that the paper's own argument is visibly wrong, but that the central claim rests on an external theorem whose precise hypotheses are not stated. Because the reader already rendered a CONDITIONAL verdict on exactly this basis, my stress-test does not change the verdict; it confirms the need to verify the [ER21] dependency before the enlargement theorem can be regarded as established.","tokens_in":68,"tokens_out":15228,"duration_ms":208133,"concrete_test":"Obtain the published or arXiv version of [ER21] and verify the precise statement of the theorem invoked at the start of §5.2. Check specifically: (a) whether it applies to stacks with a dense open properly stable locus, or only to stacks with a single properly stable point plus irreducibility; (b) whether the sequence of saturated blow-ups terminates after finitely many steps in a Deligne-Mumford stack, rather than only reducing stabilizers; and (c) whether the centers are pulled back from the good moduli space, as required for the closure argument in the induction of Theorem 5.1. If the [ER21] statement matches Theorem 5.1's hypotheses, this concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central enlargement result, Theorem 5.1, is proved in Section 5.2 by invoking, without proof or even a precise statement, the main theorem of [ER21]: 'By the main theorem in [ER21] there is a sequence of saturated blow-ups Xn → ... → X such that Xn is Deligne-Mumford.' Everything after that — the induction using extended weighted blow-ups, Lemma 5.11, and Proposition 2.16 — converts that external sequence into a line bundle LDM on an enlargement ~X whose semistable locus is proper and Deligne-Mumford. If [ER21]'s theorem requires hypotheses that are not met by the stacks considered here, Theorem 5.1 fails and with it the boundary description in Theorem 1.1 and the plane-cubics and 2n-point applications. The paper does not state the [ER21] theorem, so it does not check the match between the Abstract's stronger phrase 'any algebraic stack with a properly stable point' and Theorem 5.1's actual hypothesis of a dense open U with X ×_X U Deligne-Mumford. This is not an internal contradiction, but it is an unresolved dependency on a result whose exact content is load-bearing and unverified in the manuscript.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs compact moduli stacks of stable maps from genus-g curves to quotient stacks X=[W/G] with a projective good moduli space and a suitably dense Deligne-Mumford locus. The main new ingredient is the extended weighted blow-up, used to prove an enlargement theorem (Theorem 5.1) producing ~X with a line bundle whose semistable locus is proper and Deligne-Mumford. The quasimap theory of Section 3 then yields the compactification Q_{g,n}(~X,~X_DM,beta). Applications include fibred log-Calabi-Yau pairs, toric quotients, GIT plane cubics, and a modular proof of Hassett's conjecture on weighted pointed rational curves. The central enlargement step is conditional on an unstated theorem of [ER21].","tokens_in":54418,"tokens_out":15826,"duration_ms":152004,"significance":"If the enlargement theorem holds, this is a substantial extension of quasimap compactifications beyond the affine-quotient setting, and the extended weighted blow-up is a genuine new tool. The applications are nontrivial and are supported by explicit computations, including the automorphism-group analysis in Appendix A. I found no circularity: the enlargement is constructed from an external saturated-blow-up sequence, the main theorems do not assume their own conclusions, and there are no fitted free parameters. The main correctness risk is the unresolved dependency on [ER21].","major_comments":[{"comment":"The proof begins by invoking 'the main theorem in [ER21]' to obtain a sequence of saturated blow-ups X_n -> ... -> X with X_n Deligne-Mumford, but that theorem is never stated and its hypotheses are never checked. The paper's Theorem 5.1 assumes a dense open U such that X x_X U is Deligne-Mumford, while the abstract phrases the result as 'any algebraic stack with a properly stable point'; these formulations must be reconciled with the [ER21] result, including the notion of properly stable point in Lemma 2.4 and the openness of the stable locus in [ER21, Proposition 2.6]. Because the induction with extended weighted blow-ups, Lemma 5.11, and Proposition 2.16 converts this external sequence into the line bundle L_DM and hence into the boundary description in Theorem 1.1, the enlargement theorem is unsupported unless the [ER21] theorem is stated precisely and its hypotheses are verified. This is a load-bearing dependency, not a mere citation issue.","section":"§5.2, proof of Theorem 5.1"},{"comment":"The reduction to G = G_m^r x F is not justified as written. The proof asserts 'From [Bri15], there is a finite subgroup F < G and a surjective morphism G_m^r ⋊ F -> G with finite kernel. As G_m^r is contained in the center of G, the product is direct.' A central extension of G_m^r by a finite group is not automatically a direct product, and the existence of a semidirect-product presentation with finite kernel does not by itself imply that the F-action on G_m^r is trivial. Since the subsequent character-semistability argument uses the direct-product structure, Theorem 4.2 needs either a precise statement from [Bri15] showing that this splitting is available, or a modified argument.","section":"§4.2, proof of Theorem 4.2"}],"minor_comments":[{"comment":"The notation is mismatched: the morphism is pi: ~X -> X and p: X -> X, so the open substack should be described using (p ∘ pi)^{-1}(U), not (pi ∘ p)^{-1}(U); also U should be explicitly identified as an open substack of X.","section":"Theorem 5.1(2)"},{"comment":"There is a typo: 'stabe quasimaps' should be 'stable quasimaps'.","section":"Definition 3.16"},{"comment":"The reference [ER21] is cited with no arXiv number or version; because the main enlargement theorem depends on it, full bibliographic data and at least a precise statement of the cited theorem should be supplied.","section":"References"},{"comment":"The symbol '[Nx/Gn]' should presumably be '[N_x/G_x]'.","section":"Definition 2.20"},{"comment":"In Step 2, the sentence about pi inducing an isomorphism of good moduli spaces uses the root-stack descent from ~CCY_{2n} to CCY_{2n}; the descent is plausible from Proposition A.7 but is not spelled out, so a clarifying sentence would remove ambiguity.","section":"Appendix A, proof of Theorem A.4"},{"comment":"The phrase 'any algebraic stack with a properly stable point' promises more than the technical hypothesis of Theorem 5.1; aligning the abstract with the actual statement would prevent a misleading first impression.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle to acceptance is the dependence on [ER21]. If the authors can state the [ER21] theorem precisely and verify its hypotheses for the stacks considered, I expect the core results to be sound; otherwise the paper would need to prove the needed enlargement result or weaken the main theorem. I recommend major revision rather than rejection because the architecture is coherent and the applications are valuable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis is a serious paper. The main theorem is real progress: it gives a proper DM stack compactifying maps from curves to a quotient stack [W/G] with projective good moduli space, provided the target has a dense Deligne-Mumford locus. The new birational operation (extended weighted blow-up) and the enlargement theorem are genuine contributions, and the quasimap framework is adapted carefully to this relative setting. The applications are concrete: fibered log CY pairs, plane cubics, and a modular proof of Hassett's conjecture. Credit where due: the architecture of the proof is coherent, the proofs are detailed, and the examples are worked out.\n\nThe soft spots are real but not fatal. First, the enlargement theorem (Theorem 5.1) depends at its start on the main theorem of [ER21], a preprint whose exact statement is never given. The authors just write 'by the main theorem in [ER21]' and proceed. Anyone refereeing this needs to check that [ER21]'s hypotheses match the paper's condition (dense open U with X ×_X U Deligne-Mumford). I believe the match is likely correct, but the paper should state the theorem it is using and verify the hypotheses explicitly. Given that [ER21] appears to be unpublished, the authors should either include the needed statement or prove the special case.\n\nSecond, the abstract overclaims. It says any stack 'with a properly stable point' can be enlarged, but Theorem 5.1 requires a dense open U over which the stack is DM. If the stack is reducible, a single properly stable point need not be open-dense, so the abstract's phrasing is stronger than what is proved. Remark 5.3 notes this works for irreducible X, but that caveat is absent from the abstract.\n\nThird, a few arguments in Section 3 are abbreviated, particularly the boundedness and obstruction theory parts. They follow known routes (Cheong-Ciocan-Fontanine-Kim-Maulik, Halpern-Leistner-Herrero), so I would call this minor — but a referee should ask for the missing details where the adaptation is non-obvious.\n\nOverall: this deserves a serious referee, and I expect the referee to return substantive comments but not a rejection. The paper is for people working on moduli of varieties of intermediate Kodaira dimension, quasimap theory, and GIT compactifications. My recommendation: send it to peer review. The main things to fix are the abstract and the explicit statement of the [ER21] input.","headline":"A substantial compactification theorem for maps to quotient stacks, with a real but checkable dependency on an external preprint that the authors should spell out.","tokens_in":54998,"tokens_out":4220,"would_cite":true,"duration_ms":41201,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14D23","14D20","14L24","14H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the moduli of maps from smooth curves to a quotient stack can always be compactified, provided the stack has a projective good moduli space and a dense Deligne-Mumford locus, by enlarging the target with extended…","keywords":["stable quasimaps","quotient stacks","extended weighted blow-up","good moduli space","twisted curves","GIT compactification","moduli of maps","Hassett conjecture"],"falsifier":"An explicit family of smooth genus-g maps to a quotient stack X=[W/G] over the punctured spectrum of a DVR whose limit, after every finite sequence of extended weighted blow-ups, still has no unique stable quasimap limit would refute Theorem 1.1. A concrete arena is the GIT compactification of plane cubics treated in Section 6: a pencil of cubics with twelve nodal fibers whose central fiber acquires a cusp or worse would violate the claimed boundary description.","tokens_in":53955,"feed_emoji":"📐","tokens_out":12750,"duration_ms":102700,"temperature":0.7,"pith_summary":"The paper establishes that the moduli stack of maps from smooth genus-g curves to a quotient stack X=[W/G]—where G is reductive, X has a projective good moduli space, and a dense open substack of X is Deligne-Mumford—admits a proper Deligne-Mumford compactification. The strategy is to enlarge the target: a new birational operation, the extended weighted blow-up, produces a larger quotient stack X~ with the same good moduli space and containing a proper Deligne-Mumford open substack. Stable quasimaps from twisted curves to X~ then form a proper Deligne-Mumford stack that generically parametrizes the original smooth-curve maps to X. The same machinery yields compact moduli for GIT compactifications of binary forms, 2n-marked rational curves, and plane cubics, and for certain fibered log Calabi-Yau pairs.","feed_headline":"A new blow-up compactifies stable maps to quotient stacks","feed_subtitle":"One birational operation enlarges any such stack until an open proper Deligne-Mumford substack appears.","key_machinery":"The central object is the extended weighted blow-up of an algebraic stack X along a weighted ideal sequence I_•, defined as the quotient $[\\mathrm{Spec}_{\\mathcal{X}}(\\bigoplus_n I_n)/\\mathbb{G}_m]$—the Gm-quotient of the deformation of X to the weighted normal cone determined by I_•. It is a birational transformation that keeps the good moduli space unchanged, contains X as a dense open substack, and contains the ordinary weighted blow-up as an open substack. Iterating extended weighted blow-ups along the centers of a saturation sequence produces the enlargement X~, and a relative-GIT comparison (Proposition 2.16) shows that the semistable loci of the intermediate steps assemble into a proper Deligne-Mumford open substack of X~. The compactification is then built by combining twisted-curve theory with quasimap theory: stable quasimaps from twisted curves to X~ are defined by a stability condition on the line bundle ω_C(Σ p_i)⊗f^*L^⊗3, and the fixed class β together with quasimap boundedness makes the resulting stack Q_g(X~, X~_DM, β) proper and Deligne-Mumford.","core_discovery":"The paper's main theorem states that for every quotient stack X=[W/G] with G reductive, with a projective good moduli space X→X, and with a dense open U⊂X such that X×_X U is Deligne-Mumford, there is a proper Deligne-Mumford stack Q_g(X~, X~_DM, β) that generically parametrizes morphisms φ:C→X of class β from smooth genus-g curves and whose boundary parametrizes stable quasimaps from twisted curves to an enlargement X~ of X. The enlargement is constructed in Theorem 5.1 by an explicit birational method: starting from a sequence of saturated blow-ups of X (whose existence is imported from the literature), each step is realized as an extended weighted blow-up, and a relative-GIT comparison identifies a line bundle on the resulting stack whose semistable locus is proper and Deligne-Mumford. When X already contains a dense open proper Deligne-Mumford substack—as happens for the moduli of boundary-polarized Calabi-Yau pairs and for torus quotients of Deligne-Mumford stacks—the enlargement is unnecessary and the compactification is built directly from X. Applications include compact moduli of fibered log Calabi-Yau pairs of Kodaira dimension one and compactifications of maps to GIT quotients of binary forms, 2n-marked rational curves, and plane cubics.","pith_inferences":["Because the enlargement is built explicitly from weighted ideal sequences, one could in principle compute the boundary stratification of Q_g for a given GIT quotient by tracking the saturation algorithm step by step.","The same strategy should apply to stacks with several properly stable points or with non-trivial stabilizers at the polystable locus, as long as the centers of the extended blow-ups are chosen compatibly; the plane-cubics example shows the stabilizer group of the polystable point controls how many steps are needed.","A testable extension is allowing colliding marked points with Hassett-type weights; the authors conjecture (Remark 3.25) that their arguments go through, which would unify quasimap stability with weighted pointed stability.","If future work produces saturation sequences that are canonical or minimal, Theorem 5.1 would turn any such sequence into a compact moduli of maps, effectively reducing the compactification problem to the existence of saturation sequences."],"forward_implications":["For every quotient stack X with a projective good moduli space and a dense Deligne-Mumford locus, the space of maps from smooth curves to X has a proper Deligne-Mumford compactification Q_g(X~, X~_DM, β).","When X already contains a dense open proper Deligne-Mumford substack, no enlargement is needed; this covers moduli of boundary-polarized Calabi-Yau pairs and torus quotients of Deligne-Mumford stacks.","The boundary of the compactification parametrizes stable quasimaps from twisted curves to the enlarged target, so degenerations of maps are described by modular objects rather than by an abstract completion.","In the smoothable case (X lci and X_DM smooth), the moduli stack carries a perfect obstruction theory, allowing virtual fundamental class counts.","The appendix gives a modular proof of Hassett's conjecture: the morphism from a stack of twisted conics to the GIT stack of 2n unordered points on P^1 is an extended weighted blow-up, recovering the known weighted blow-up on coarse moduli spaces."],"supporting_citations":[{"why":"Supplies the sequence of saturated blow-ups that the enlargement construction in Theorem 5.1 starts from.","marker":"[ER21]"},{"why":"Provides the boundedness theorem for quasimaps from a fixed curve that Section 3 extends to families.","marker":"[CFKM14]"},{"why":"Gives the orbifold quasimap theory whose stability and compactness arguments are generalized to arbitrary quotient stacks.","marker":"[CCFK15]"},{"why":"Supplies the twisted-curve and twisted-stable-map machinery used to define the boundary of Q_g.","marker":"[AV02]"},{"why":"Establishes the good-moduli-space and relative-GIT foundations used throughout, including the semistable-locus comparison in Proposition 2.16.","marker":"[Alp13]"},{"why":"Defines weighted blow-ups and weighted ideal sequences that the new extended weighted blow-ups generalize.","marker":"[QR21]"},{"why":"Provides the moduli of boundary-polarized Calabi-Yau pairs used in the Kodaira-dimension-one application (Theorem 1.2).","marker":"[BL24]"},{"why":"Defines the weighted pointed stable curves whose GIT quotient is re-derived modularly in the appendix (Hassett's conjecture).","marker":"[Has03]"}],"fun_headline_variants":["Weighted blow-ups compactify maps to quotient stacks","Extended blow-up yields proper DM compactifications","New blow-up technique compacts stable map moduli","Blow-up method builds compact moduli for stack maps","Weighted blow-up enlarges stacks to proper DM substacks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on an imported theorem, not proved in this paper, that every algebraic stack with a good moduli space and a properly stable point admits a finite sequence of saturated blow-ups ending in a Deligne-Mumford stack; if that sequence does not exist for some target X, the enlargement and the compactification are not obtained.","fun_headline_variants_meta":{"raw":{"variants":["Weighted blow-ups compactify maps to quotient stacks","Extended blow-up yields proper DM compactifications","New blow-up technique compacts stable map moduli","Blow-up method builds compact moduli for stack maps","Weighted blow-up enlarges stacks to proper DM substacks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00019,"raw_usage":{"total_tokens":1399,"prompt_tokens":1068,"completion_tokens":331,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":684,"completion_tokens_details":{"reasoning_tokens":254}},"tokens_in":684,"tokens_out":331,"duration_ms":3757,"temperature":1.0,"reasoning_tokens":254,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:30:31.901256+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"An explicit family of smooth genus-g maps to a quotient stack X=[W/G] over the punctured spectrum of a DVR whose limit, after every finite sequence of extended weighted blow-ups, still has no unique stable quasimap limit would refute Theorem 1.1. A concrete arena is the GIT compactification of plane cubics treated in Section 6: a pencil of cubics with twelve nodal fibers whose central fiber acquires a cusp or worse would violate the claimed boundary description.","supporting_citations":[],"review_version":1}