{"id":"6939fd2d-c6a8-4082-808c-4280727f37a3","arxiv_id":"2412.16295","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For any smooth projective toric variety, a contraction morphism from stable maps to quasimaps is constructed on a closed substack, and is proven surjective when the target is Fano.","lead":"The paper builds a bridge between two popular ways of compactifying spaces of curves mapping into toric varieties: stable maps and quasimaps. It shows that for Fano toric targets every quasimap can be obtained from a stable map by a natural contraction process, which may make Gromov-Witten and quasimap invariants comparable by geometry alone.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Construction 5.1.2 depends on an auxiliary epic embedding, and Proposition 5.2.1 establishes independence only at the level of closed points, so the stack structure of M^c and the morphism c_X are not yet shown to be canonical.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing issue: Construction 5.1.2 chooses an epic embedding iota, and Proposition 5.2.1 proves only independence of closed points and of the pointwise formula, not independence of the stack M^c or of the morphism c_X as a morphism of stacks. This is a genuine gap in the paper's presentation because the central object is called 'the contraction morphism of X' and is used in Theorem 6.0.1 without tracking the auxiliary choice. The paper itself flags this limitation in the paragraph after Construction 5.1.2, so the concern is not manufactured. The rest of the argument appears coherent: the degree of a basepoint is well-defined, the examples check the definitions, Corollary 4.3.4 supplies the necessary closed embedding of quasimap spaces, and the surjectivity proof, though compressed, seems to follow from grafting and induction on the anticanonical length. I do not see an internal contradiction or a counterexample to the main construction; the issue is a missing stack-level independence proof. Therefore the appropriate verdict remains CONDITIONAL, with the condition being that the auxiliary-choice dependence is resolved. Since my concern does not move the reader's verdict, I mark the recommendation as UNCHANGED.","tokens_in":31450,"tokens_out":25955,"duration_ms":254325,"concrete_test":"Take X = Bl_0 P^2 and compare the two epic embeddings iota and i of Example 4.2.6, namely into P^2 × P^1 and P^1 × P^4. For a suitable family of stable maps, e.g. the family in Example 5.2.2 with class 2L, compute the ideal sheaf of the fiber-product substack in M_{0,2}(Bl_0 P^2, 2L) for each embedding by testing first-order deformations over Spec C[epsilon]/(epsilon^2). If both ideal sheaves agree on this non-reduced test family, this supports stack-level independence; if the induced equations differ, the construction is genuinely choice-dependent and Proposition 5.2.1 must be upgraded to a family-level statement before M^c and c_X are called canonical.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Construction 5.1.2, the closed substack M^c_{g,n}(X,beta) is defined as the fiber product Q_{g,n}(X,beta) ×_{Q_{g,n}(P,iota_*beta)} M_{g,n}(P,iota_*beta), so it depends on the chosen epic closed embedding iota into a product of projective spaces. Proposition 5.2.1 explicitly addresses only 'the description of the closed points of M^c' and the pointwise formula for c_X, and the text acknowledges that the definition 'a priori depends on the chosen epic closed embedding iota'. Two closed substacks of a stack can have the same closed points while having different non-reduced structures or different universal families, so equal C-points does not imply a natural isomorphism of the fiber-product stacks. Because Q(iota) is a closed immersion, M^c_iota -> M_{g,n}(X,beta) is a closed immersion, but without a proof that the corresponding ideal sheaves or universal factorizations agree, the phrase 'the contraction morphism of X' is not justified. This is load-bearing for the canonicality claim in the abstract and for any future use of c_X in virtual-class comparisons, although it does not overturn the existence of a contraction morphism for each fixed choice of iota.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for every smooth projective toric variety X, a closed substack M^c_{g,n}(X,β) of the stable maps stack and a morphism c_X from it to the toric quasimaps stack Q_{g,n}(X,β), extending the identity on maps from smooth curves. The construction is based on a new invariant: the degree β_x of a quasimap at a basepoint, defined combinatorially and characterised by a universal twisting property. The paper proves that a quasimap is determined by its regular extension and the degrees at its basepoints, that pushforwards along epic closed embeddings of quasimap spaces are closed embeddings, and that for Fano X the contraction morphism is surjective. The motivation is a geometric comparison between Gromov–Witten and quasimap invariants for toric targets.","tokens_in":31634,"tokens_out":14069,"duration_ms":128717,"significance":"If correct, the paper provides a genuinely useful comparison morphism for all smooth projective toric varieties, generalising the contraction map for projective spaces and for one parametrized component. The basepoint-degree invariant is a valuable new tool: it explains the difference between quasimap degree and regular-map degree, recovers the length of a basepoint, and controls injectivity of functoriality maps between quasimap spaces. Theorem C, that Q(ι) is a closed embedding when ι is epic, is a clean structural result. The surjectivity theorem for Fano targets is a strong statement that could support future virtual-class comparison arguments. The main limitations are that the contraction construction is only proved to be independent of the auxiliary embedding at the level of closed points, and that the surjectivity proof contains a compressed induction step that needs to be made rigorous.","major_comments":[{"comment":"Construction 5.1.2 defines M^c_{g,n}(X,β) and c_X using an auxiliary epic closed embedding ι into a product of projective spaces. The text acknowledges the a priori dependence on ι, but Proposition 5.2.1 only proves that the closed points of M^c and the pointwise formula for c_X are independent of ι. Since M^c is a closed substack of M_{g,n}(X,β), having the same closed points does not imply that the stack structures (nilpotent structure, universal family, or the closed immersion j) agree for different choices of ι. Thus the phrase 'the contraction morphism of X' is not yet justified as a canonical object. For each fixed ι the existence of a contraction morphism is not in question, but either a stack-level independence statement must be proved or the paper should systematically state the results for a chosen epic embedding.","section":"Section 5.1, after Construction 5.1.2"},{"comment":"After reducing to one basepoint, the proof says 'by restricting ourselves to the irreducible component containing the basepoint x, we can assume that C is irreducible.' This reduction is not justified: restricting a quasimap to one irreducible component discards the other components, their nodes, and any markings on them, and a stable map constructed on that component does not automatically extend to a stable map on the original curve with c_X(f)=q. The multi-component case needs a separate argument — for example by grafting rational tails onto the remaining components and carefully tracking the total class, the markings, and stability — rather than a one-sentence reduction.","section":"Section 6.3, proof of Theorem 6.0.1"},{"comment":"The introduction defines M^c_{g,n}(X,β) as Q_{g,n}(X,β) ×_{Q_{g,n}(P,ι_*β)} M_{g,n}(P,ι_*β). This is not the object constructed in Section 5.1, where the Cartesian square is taken with M_{g,n}(X,β) in the upper-left corner, i.e. M^c = M_{g,n}(X,β) ×_{Q_{g,n}(P,ι_*β)} Q_{g,n}(X,β). The introductory formula would not define a closed substack of M_{g,n}(X,β). Please correct the introductory definition so that it agrees with Construction 5.1.2.","section":"Section 1.1, definition of M^c"},{"comment":"Corollary 4.3.4 concludes that Q(ι) is a closed embedding because it is proper and a monomorphism. The proof of monomorphism cites Theorem 4.3.3, which explicitly describes fibres as sets of closed points. Since the statement concerns algebraic stacks over C, the argument should either spell out that the same description applies to all geometric points, or cite a criterion that makes closed-point injectivity plus properness sufficient for a monomorphism in this setting. As written, the step is slightly too compressed.","section":"Section 4.3, Corollary 4.3.4"}],"minor_comments":[{"comment":"The sentence 'we came up with Definition 3.2.10 while studying...' is informal and out of place in a research paper; it should be removed or moved to an introductory remark.","section":"Section 3.2, after Definition 3.2.10"},{"comment":"There is a typo: 'existance' should be 'existence'.","section":"Proof of Proposition 3.2.9"},{"comment":"The word 'instrinsic' should be 'intrinsic'.","section":"Remark 5.2.3"},{"comment":"The typeset diagram is hard to parse. Adding explicit labels such as j, c_X, M(ι), cP, and Q(ι) on the actual arrows, or writing the fibre product in a displayed equation, would improve readability and prevent ambiguity.","section":"Diagram (24) and the Cartesian diagram in Construction 5.1.2"},{"comment":"The phrase 'As a sanity check for Proposition 5.2.1' is informal; consider replacing it with 'As a verification of Proposition 5.2.1'.","section":"Example 5.2.2"}],"recommendation":"major_revision","confidential_remarks":"The paper contains useful new constructions and fits the journal's scope. I would be willing to review a revised version. The two issues that block acceptance are the embedding-dependence of the contraction stack (only closed-point independence is proved) and the unjustified reduction to an irreducible source curve in the proof of surjectivity. Both appear fixable within the manuscript's framework. The introductory definitional inconsistency should also be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper. The new thing is a construction, for every smooth projective toric variety X, of a closed substack of stable maps and a morphism c_X from it to quasimaps, with surjectivity when X is Fano. The main engine is a new invariant, the degree of a basepoint, which measures the discrepancy between quasimap degree and regular extension degree and recovers the CFKM length. That invariant is well-defined, has good properties, and is used to prove a nice criterion: along an epic embedding (injective on A1), the induced map on quasimap spaces is a closed embedding. The examples, especially the Segre one, are genuinely illuminating.\n\nThe proof of surjectivity is an induction on anticanonical length with grafting of rational tails. I did not mechanically check every Stacks Project citation, but the argument is coherent and the difficult part—excluding loops of basepoints—is addressed with two explicit lemmas. The paper is honest about its limitations: it states that the construction a priori depends on the auxiliary epic embedding, and that only independence of closed points is proven.\n\nThe soft spots are real but not fatal. First, the definition of M^c in the Introduction writes it as a fiber product of Q(X) and M(P) over Q(P); that stack does not map to M(X). The correct fiber product, used in Construction 5.1.2, is M(X) ×_{Q(P)} Q(X). This is a typo, but in a paper about a new moduli stack, an inconsistent definition in the introduction is a stumbling block. Second, the top-level claim that c_X is 'the contraction morphism of X' is stronger than what is proved: Proposition 5.2.1 shows the closed points and the pointwise formula do not depend on the embedding, but not that the stack structures agree. If the paper is meant to support virtual-class comparison, that stack-level canonicality needs proof; otherwise the theorems should be phrased as 'for a choice of epic embedding.' Third, the induction in Theorem 6.0.1 is compressed at the multi-basepoint step; a reader has to fill in why grafting one basepoint at a time preserves the induction invariant.\n\nNone of this shakes the main construction. The paper deserves a serious referee, and my recommendation would be to send it to review with a request to fix the intro definition, clarify the canonicality claim, and expand the induction. It is worth reading if you work on moduli or wall-crossing for toric targets.","headline":"A genuinely new toric contraction morphism with a useful basepoint invariant; the main theorem is believable and the flaws are presentation-level, so it deserves a real referee.","tokens_in":32220,"tokens_out":4499,"would_cite":true,"duration_ms":38826,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14N35","14M25","14D23"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every smooth projective toric variety, the paper constructs a contraction morphism from a closed substack of stable maps to the quasimap space, and proves it is surjective when the target is Fano.","keywords":["toric varieties","quasimaps","stable maps","contraction morphism","basepoint degree","moduli of curves","curve-counting invariants","Fano toric varieties"],"falsifier":"Compare, on the nonreduced family $\\operatorname{Spec} \\mathbb{C}[\\varepsilon]/\\varepsilon^2$, the two fiber-product substacks obtained from the two epic embeddings of $\\mathrm{Bl}_0\\mathbb{P}^2$ written in Example 4.2.6; the paper proves coincidence only on closed points, so any difference in the stacks or in $c_X$ would show the construction is choice-dependent. Separately, run the grafting algorithm of Section 6.2 on the quasimap of Example 5.2.4: if at any stage the required sections $t_\\rho$ on the grafted $\\mathbb{P}^1$ cannot be chosen, surjectivity for Fano targets would fail.","tokens_in":31166,"feed_emoji":"🔁","tokens_out":12651,"duration_ms":102575,"temperature":0.7,"pith_summary":"The paper connects two compactifications of the space of maps from curves to a smooth projective toric variety: stable maps and stable quasimaps. For any smooth projective toric variety $X$ it constructs a closed substack of the stable-maps moduli space and a contraction morphism from that substack to the quasimap moduli space, agreeing with the identity on maps from smooth curves. When $X$ is Fano, the contraction morphism is proved to be surjective, so every stable quasimap is the contraction of an actual stable map. The new invariant that makes this work is the degree of a quasimap at a basepoint: an effective curve class attached to each basepoint that accounts exactly for the gap between the quasimap degree and the degree of its regular extension. This gives a concrete geometric mechanism for comparing the two enumerative invariants of the spaces.","feed_headline":"Every Fano toric quasimap is a contraction of a stable map","feed_subtitle":"The degree gap between a quasimap and its regular map is carried by basepoints, making both spaces comparable.","key_machinery":"The central object is the degree of a basepoint (Definition 3.2.10): the unique effective curve class $\\beta_x \\in A_1(X)$ such that twisting the quasimap's line-bundle-section data by $-\\beta_x$ at $x$ makes $x$ a regular point. This class is built combinatorially from the vanishing orders of the sections against maximal cones of the toric fan, and it satisfies the identity $\\beta = \\beta_{\\mathrm{reg}} + \\sum_{x\\in B} \\beta_x$. It supplies two mechanisms: it is the invariant that distinguishes quasimaps with the same regular extension, and it is the data used to graft a rational curve at a basepoint in the proof of surjectivity. The contraction morphism itself is assembled by embedding $X$ into a product of projective spaces through an epic closed embedding and restricting the projective-space contraction; on a stable map it contracts rational tails while twisting the remaining sections by the tail degrees.","core_discovery":"The paper's central claim is that the contraction morphism, previously available for projective space, can be constructed for every smooth projective toric variety. One chooses an epic closed embedding of $X$ into a product of projective spaces, meaning an embedding whose induced map on curve classes is injective; Corollary 4.3.4 shows such embeddings make the quasimap space of $X$ a closed substack of the quasimap space of the product. Pulling back the product's contraction morphism through this closed embedding defines $c_X$ on a closed substack $M^c_{g,n}(X,\\beta)$ of the stable maps stack (Construction 5.1.2). On points, $c_X$ contracts each rational tail of a stable map and twists the remaining line-bundle sections by the tail's degree; Proposition 5.2.1 characterizes which stable maps lie in the substack. Theorem 6.0.1 proves surjectivity for Fano $X$ by grafting rational curves onto the basepoints of any quasimap until a stable map is obtained. Along the way the paper proves a quasimap is determined by its regular extension, its basepoints, and the degree of each basepoint.","pith_inferences":["If the embedding-independence issue is resolved, the same construction would give a canonical contraction morphism whose pointwise formula could be used to compare virtual fundamental classes of the two moduli spaces without wall-crossing.","The surjectivity proof reads as a terminating combinatorial algorithm on the fan: resolve basepoints by successively grafting rational curves, with the Fano condition guaranteeing the required section choices exist; running it on non-Fano toric surfaces should locate precisely where the process can loop forever.","The degree of a basepoint is finer than the previously known length invariant, so it may distinguish quasimaps that have the same length; the two quasimaps in Example 4.1.1 are a natural test case."],"forward_implications":["For smooth Fano toric $X$, every stable quasimap of class $\\beta$ is the image under $c_X$ of a stable map of class $\\beta$.","A quasimap is uniquely determined by its regular extension, the set of its basepoints, and the degree of each basepoint (Corollary 3.4.2).","The degree of a basepoint explains the difference in degree: $\\beta = \\beta_{\\mathrm{reg}} + \\sum_{x\\in B} \\beta_x$, with $\\beta_x=0$ exactly away from basepoints.","Closed embeddings between toric varieties that are injective on curve classes induce closed embeddings of quasimap spaces (Corollary 4.3.4).","If every toric boundary divisor of $X$ is numerically effective, the contraction morphism is defined on the entire stable maps space rather than only a closed substack."],"supporting_citations":[{"why":"Introduces stable toric quasimaps and their moduli stack, the object the contraction morphism targets.","marker":"[CFK10]"},{"why":"Gives the functor of points of smooth toric varieties by line bundles and sections, the language in which quasimaps and the contraction formulas are written.","marker":"[Cox95]"},{"why":"Supplies the toric facts on Picard groups, nef and Mori cones, effectivity, and the anticanonical embedding used throughout.","marker":"[CLS11]"},{"why":"Contains the counterexample showing quasimap spaces do not embed along arbitrary closed embeddings, motivating the epic-embedding condition.","marker":"[BN21]"},{"why":"Introduces the contraction morphism for projective space and the comparison that this paper generalizes to toric varieties.","marker":"[MOP11]"},{"why":"Defines the length of a basepoint, which the paper's new degree of a basepoint recovers in Lemma 3.5.1.","marker":"[CFKM14]"},{"why":"Provides the family version of the contraction morphism for projective space and context for where such a morphism fails to extend.","marker":"[PR03]"},{"why":"Supplies the stack-theoretic lemmas used to prove that quasimap spaces embed under epic embeddings and that the contraction morphism is surjective on closed points.","marker":"[Sta22]"}],"fun_headline_variants":["Toric quasimaps contracted from stable maps","For Fano toric varieties, every quasimap lifts to a stable map","Surjective contraction from stable maps to Fano toric quasimaps","All Fano toric quasimaps arise from stable map contractions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything hangs on one auxiliary choice: an embedding of $X$ into a product of projective spaces; the paper proves the points of the resulting substack are independent of that choice, but not that the substack as a whole is, so the uniqueness of \"the\" contraction morphism is not yet established.","fun_headline_variants_meta":{"raw":{"variants":["Toric quasimaps contracted from stable maps","For Fano toric varieties, every quasimap lifts to a stable map","Surjective contraction from stable maps to Fano toric quasimaps","All Fano toric quasimaps arise from stable map contractions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1436,"prompt_tokens":880,"completion_tokens":556,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":496,"completion_tokens_details":{"reasoning_tokens":481}},"tokens_in":496,"tokens_out":556,"duration_ms":5056,"temperature":1.0,"reasoning_tokens":481,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:45:22.426679+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compare, on the nonreduced family $\\operatorname{Spec} \\mathbb{C}[\\varepsilon]/\\varepsilon^2$, the two fiber-product substacks obtained from the two epic embeddings of $\\mathrm{Bl}_0\\mathbb{P}^2$ written in Example 4.2.6; the paper proves coincidence only on closed points, so any difference in the stacks or in $c_X$ would show the construction is choice-dependent. Separately, run the grafting algorithm of Section 6.2 on the quasimap of Example 5.2.4: if at any stage the required sections $t_\\rho$ on the grafted $\\mathbb{P}^1$ cannot be chosen, surjectivity for Fano targets would fail.","supporting_citations":[],"review_version":1}