{"id":"f85a6a92-19e3-4b0e-96b6-dcb7324582ea","arxiv_id":"2412.02817","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Tropical psi classes are shown to be the tropicalization of algebraic psi classes for families that are tropicalizable and have enough affine functions near the section.","lead":"This paper proves that, when a family of curves degenerates enough near a marked point, the tropical version of the psi class equals the tropicalization of the algebraic psi class. It resolves a question from earlier work and introduces a general way to give tropicalizations of toroidal varieties an affine structure.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Corollary 5.11 drops completeness: Proposition 5.10 requires a complete toroidal base, so Trop(ψ_i) is undefined for non-proper B.","rationale":"The reader identified the tropicalizable condition and the Aff_C(−s_i) line-bundle hypothesis as the fragile assumptions; I agree these are restrictive, but they are explicitly stated conditions of the theorem, so they are not gaps in the argument. A more concrete and load-bearing issue is that Corollary 5.11 invokes Proposition 5.10 without carrying over its completeness hypothesis. Definition 5.6 defines tropicalization of a Chow class via intersection numbers with boundary strata; these numbers require a proper/complete ambient variety. The family in Definition 3.18 has no such condition on B, and the introductory statement of the main theorem repeats the omission. Consequently, the final equality of cycles is not justified in the stated generality. The paper is otherwise careful: Theorem 5.5 gives an isomorphism of tropical line bundles, and the extended example provides strong supporting evidence in a proper case. The fix is small—add completeness to the theorem statements or prove a local/stacky version—so the appropriate action is conditional acceptance rather than rejection. I would ask the authors to clarify the hypothesis and confirm that all intended applications fall under it.","tokens_in":37566,"tokens_out":14310,"duration_ms":158934,"concrete_test":"Trace the proof of Corollary 5.11 and check whether Definition 3.18 or any intervening statement implies B is complete. If no such implication exists, modify the statement to read: 'Let C→B be a tropicalizable family over a complete toroidal variety B' and verify that all applications (Section 6, the M_g,n setting) satisfy this. Alternatively, construct an explicit non-proper example—e.g., B=A^1 with the trivial family C=B×P^1 and two sections—and attempt to compute both sides of Trop(ψ_i)=ψtrop_i; if the left side is not finite or not defined, the counterexample confirms the gap.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central cycle-level claim, Corollary 5.11, states that for any tropicalizable family C→B with Aff_C(−s_i) a tropical line bundle, Trop(ψ_i)=ψtrop_i. Its proof is: 'Combine Theorem 5.5 and Proposition 5.10.' But Proposition 5.10 begins 'Let X be a complete toroidal variety...' and Definition 5.6 defines Trop_X(c) by the intersection number ∫_X c·[V(σ)], which is only finite when X is complete (or at least proper over the base field). Definition 3.18 of a tropicalizable family does not require B to be complete, and neither Corollary 5.11 nor the statement in Section 1.2 adds such a hypothesis. Thus, for families over non-proper toroidal bases—for example, an open subset of M_g,n before compactification, or an affine base—the right-hand side of the equality is not a well-defined tropical cycle on the full extended cone complex. The paper's main examples use proper spaces (admissible covers, compactified moduli spaces) and therefore satisfy the missing hypothesis, but the theorem as written is over-stated. This is not a fatal flaw in the geometric ideas; it is a precise gap in the statement and proof of the headline theorem that can likely be fixed by adding a properness/completeness hypothesis or by formulating a local statement for the compactified family.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a notion of tropicalization for toroidal embeddings that records not only the extended cone complex but also a sheaf of affine functions, defined by requiring that certain piecewise linear functions correspond to line bundles that trivialize on strata. This makes tropicalization a functor to the category of tropical spaces, invariant under logarithmic modifications. The main result is a comparison theorem: for a tropicalizable family of stable marked curves C → B whose tropicalization admits enough affine functions near the i-th section, the tropicalization of the algebraic cotangent line bundle L_i is isomorphic to the tropical cotangent line bundle L_trop_i of [CGM22], and consequently Trop(ψ_i) = ψ_trop_i. The paper also contains an extended example in genus one, computing tropical ψ classes on a space of admissible covers and verifying consistency with algebraic pushforward relations.","tokens_in":37881,"tokens_out":5278,"duration_ms":54319,"significance":"If the comparison theorem holds as stated, it resolves the motivating question of whether the combinatorial tropical ψ classes of [CGM22] are actual tropicalizations of algebraic ψ classes, under explicit and clearly stated hypotheses. The paper's main strengths are its careful conditional formulation, its functorial and modification-invariant definition of tropicalization, the detailed treatment of line bundles and affine structures, and a substantial worked example connecting the general theory to concrete computations in tropical admissible covers. The extended example is a genuine verification that is independent of the abstract formalism and adds credibility to the main claims. The paper does not rely on black-box software or unchecked computations; the arguments are presented in enough detail to be followed, with a small number of delegated 'immediate' checks.","major_comments":[{"comment":"The statement of the headline comparison theorem omits a completeness/properness hypothesis on the base B, but its proof invokes Proposition 5.10, which begins 'Let X be a complete toroidal variety.' Definition 5.6 defines Trop_X(c) via the intersection number ∫_X c·[V(σ)], which is finite only when X is complete (or at least proper over the base field). Definition 3.18 of a tropicalizable family does not require B to be complete, and neither Corollary 5.11 nor the claim in Section 1.2 adds such a hypothesis. Thus for non-proper toroidal bases—for example an open subset of M_{g,n} or an affine base—the term Trop(ψ_i) on the left-hand side of Trop(ψ_i)=ψ_trop_i is not defined as a tropical cycle on the full extended cone complex. The paper's main examples use proper spaces and are unaffected, but the theorem as written is over-stated. Please add a completeness hypothesis to Corollary 5.11 and to the corresponding statement in the introduction and in Theorem F, or alternatively formulate and prove a local statement for proper families and explain precisely how the non-proper case is to be handled.","section":"Corollary 5.11; Section 1.2"}],"minor_comments":[{"comment":"The proof refers forward to Corollary 5.8 for the vanishing of the intersection product ϕ·[Σ^σ_X]. Since Corollary 5.8 is proved independently of Proposition 3.16, this is not circular, but the forward reference should be explicitly flagged (or the statements reordered) to avoid the appearance of a circular argument.","section":"Proposition 3.16"},{"comment":"In Cases 2 and 3, the text says the computations are 'very similar' and lists results without derivation. Since these examples are the main illustration of the difference between being a tropical line bundle on the cone complex and on the extended cone complex, it would be helpful to include the affine-function lists for the open sets U0y, Uxy, and U∞y in Case 2 and for U0y in Case 3, or to provide a table with the relevant triviality conditions.","section":"Example 4.3"},{"comment":"The notation Trop(ψ_i) in Corollary 5.11 is not explicitly defined; once completeness of B is added as a hypothesis, the authors should state that Trop(ψ_i) means Trop_B(ψ_i) as in Definition 5.6, applied to the algebraic class c1(L_i) on B.","section":"Definition 5.6 and Corollary 5.11"},{"comment":"The claim that one may remove self-intersections 'by subdividing barycentrically the generalized cone complex' could use a brief clarification of why the subdivision can be chosen combinatorially, and why the resulting affine structure is independent of the choice by Proposition 3.8.","section":"Remark 3.9"}],"recommendation":"major_revision","confidential_remarks":"The completeness gap in Corollary 5.11 is a genuine over-statement of the main theorem, but it appears to be easily fixable by adding a properness/completeness hypothesis or by stating a local version. The mathematical core of the paper is sound, and the extended example is a strong positive feature. I would encourage the editors to request the revision rather than reject, as the issue is local to the statement and does not affect the main geometric ideas."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a serious, well-written paper that delivers a conditional positive answer to the CGM22 question, and it introduces a tropicalization functor with affine structures that is worth having regardless. I would send it to a good referee.\n\nWhat's genuinely new: the affine structure on tropicalizations defined through line-bundle trivialization (Definition 3.1), the functoriality and log-modification invariance (Propositions 3.5 and 3.8), and the comparison theorem (Theorem 5.5/Corollary 5.11) identifying the tropicalization of the algebraic cotangent line bundle with the CGM22 tropical psi bundle whenever the relevant Aff torsor is a tropical line bundle. The genus-zero verification is a necessary check, and it is done cleanly. The extended admissible-cover example in Section 6 is substantial: it computes actual tropical cycles and checks compatibility with two different forgetful maps. That is real work, and it pays off.\n\nNow the soft spots. The most concrete one is the missing completeness hypothesis in Corollary 5.11. Proposition 5.10, which the proof invokes, begins 'Let X be a complete toroidal variety,' and Definition 5.6 defines Trop_X(c) by integrating c·[V(σ)] over X—only finite when X is proper/complete. Corollary 5.11 and the claim in Section 1.2 state the equality for 'a tropicalizable family' without requiring B to be complete. So for families over non-proper toroidal bases, the right-hand side is not even defined. This is a precise statement-level gap, not a flaw in the geometric idea; adding 'B complete' or restricting to a compactified family fixes it. The examples in the paper all live over proper bases (admissible covers, compactified moduli), so the computations are safe.\n\nI also flagged a few places where computations are delegated to 'immediate' checks (Example 4.3, parts of Section 6). None of them look wrong; they are just terse. The paper leans heavily on [CGM22], but that is fair—this is a sequel and the definitions are explicitly reused. The forward reference in Proposition 3.16 to Corollary 5.8 looks odd at first, but the balancing statement is independent, so there is no circularity.\n\nWho this is for: people working on logarithmic and tropical moduli of curves, intersection theory on tropical spaces, and tautological rings. It deserves a serious referee. After the completeness hypothesis is patched, the main theorem should stand.","headline":"A conditional positive answer to the CGM22 psi-class question, with a new affine tropicalization functor; the main theorem is solid but needs a completeness hypothesis added to Corollary 5.11.","tokens_in":38363,"tokens_out":3473,"would_cite":true,"duration_ms":36017,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14T05","14A20"],"pacs":[],"model":"deepseek-v4-flash","headline":"For tropicalizable families with enough affine data near a section, the tropical psi class equals the tropicalization of the algebraic psi class.","keywords":["tropical psi classes","tropicalization","affine structures on cone complexes","toroidal embeddings","moduli of curves","tropical line bundles","logarithmic geometry","admissible covers"],"falsifier":"Recompute the tropical $\\psi_1$ class on one of the two genus-one admissible-cover families of Section 6 by evaluating the balancing condition with the affine functions pulled back from $\\mathsf{M}^{\\mathrm{trop}}_{0,5}$; the paper predicts coefficients $2/3$ and $1$ on the rays of type $\\rho_a$ and $\\rho_b$, and the pushed-forward class $12\\rho_{\\mathrm{irr}}$. A mismatch with any of these numbers would refute the comparison, while a match would confirm it in a case where the affine structure is only partially available.","tokens_in":37369,"feed_emoji":"🌴","tokens_out":12385,"duration_ms":117999,"temperature":0.7,"pith_summary":"This paper answers a question left open in tropical enumerative geometry: whether the tropical $\\psi$ classes defined combinatorially for families of tropical curves are actually tropicalizations of the algebraic $\\psi$ classes on moduli spaces of curves. The authors build a notion of tropicalization for toroidal varieties that records not just the boundary cone complex but also a sheaf of affine functions, and they show that for a tropicalizable family of stable marked curves the tropicalization of the $i$-th cotangent line bundle is the combinatorial tropical cotangent bundle, whenever that bundle is a tropical line bundle. The payoff is an equality of cycle classes, $\\mathrm{Trop}(\\psi_i)=\\psi_i^{\\mathrm{trop}}$, on the base of such a family. If the theorems are right, tropical intersection theory computes genuine algebraic $\\psi$-class intersections exactly on the families where the combinatorial affine structure is rich enough, and the paper identifies the precise condition that makes this happen.","feed_headline":"Tropical psi classes are tropicalizations of algebraic psi classes","feed_subtitle":"A new affine structure on tropical families makes combinatorial and algebraic intersection theories agree.","key_machinery":"The load-bearing construction is the affine structure on the cone complex of a toroidal variety. A strict piecewise linear function $\\phi$ on the open star of a cone $\\sigma$ is declared affine at $\\sigma$ exactly when the associated line bundle $\\mathcal{O}_{X_\\sigma}(\\phi)$ restricts trivially to the stratum $V(\\sigma)$; at a cell at infinity $\\sigma/\\tau$ one additionally requires $\\phi$ to be constant on $\\tau$. This turns tropicalization into a functor from toroidal varieties to tropical spaces, invariant under logarithmic modifications. A piecewise linear function is combinatorially principal when it agrees, on every cone, with some affine function; this is exactly what makes $\\mathrm{Aff}_{\\mathsf{C}}(-\\mathsf{s}_i)$ a tropical line bundle rather than a mere pseudo-torsor. The comparison for $\\psi$ classes then runs through the sheaf $\\mathrm{Aff}_{\\mathsf{C}}(-\\mathsf{s}_i)$: its local sections are affine functions on the finite part of the tropical curve that approach the $i$-th section with slope $-1$, and Proposition 5.2 identifies $\\mathrm{Trop}(\\mathcal{O}_{\\mathcal{C}}(\\mathsf{s}_i))$ with that pseudo-torsor, which Theorem 5.5 pulls back along the section to give the tropical cotangent bundle.","core_discovery":"On the paper's own terms, the central discovery is Theorem 5.5 and Corollary 5.11. Given a tropicalizable family $\\pi:\\mathcal{C}\\to\\mathcal{B}$ of $n$-marked stable curves with tropicalization $\\Pi:\\mathsf{C}\\to\\mathsf{B}$, and given the piecewise linear function $\\varphi_i$ on $\\mathsf{C}$ that has slope one along the ray dual to the $i$-th section and slope zero on all other rays, the condition that $\\mathrm{Aff}_{\\mathsf{C}}(-\\mathsf{s}_i)$ is a tropical line bundle—equivalently, that $\\varphi_i$ is combinatorially principal—forces the tropicalization of the algebraic cotangent line bundle $\\mathcal{L}_i$ to be isomorphic to the tropical cotangent bundle $\\mathsf{L}_i^{\\mathrm{trop}}=\\mathsf{s}_i^*\\,\\mathrm{Aff}_{\\mathsf{C}}(-\\mathsf{s}_i)$. Taking first Chern classes yields $\\mathrm{Trop}(\\psi_i)=\\psi_i^{\\mathrm{trop}}$ as tropical cycles. In words: tropical geometry sees the algebraic $\\psi$ class exactly when the family degenerates enough near the section; on that locus the combinatorial $\\psi$ classes are the tropicalizations of algebraic ones.","pith_inferences":["(Pith inference) The same argument should carry over to products of $\\psi$ classes and to tautological classes built from cotangent bundles, because the Chern-class map and tropicalization commute; the paper only states the single-class comparison.","(Pith inference) The combinatorial-principality criterion gives a working definition of which algebraic divisor classes are visible to tropical geometry: a class is tropically visible exactly when its piecewise linear representative can be made affine on every stratum by subtracting an affine function.","(Pith inference) The functoriality used in Section 6 suggests a practical recipe for higher-genus computations: pull affine functions back from genus-zero pieces of the tropicalization rather than constructing the affine sheaf from scratch, which may lower the cost of computing tautological intersections.","(Pith inference) If the affine structure is invariant under log modifications, the comparison should continue to hold after blowing up the base or total space to remove self-intersections, so the theorem is likely applicable to semistable models as well as stable ones."],"forward_implications":["On the tropicalizable locus with $\\mathrm{Aff}_{\\mathsf C}(-\\mathsf s_i)$ a torsor, $\\psi_i$ is represented by an explicit weighted tropical cycle on the base, so algebraic $\\psi$-class intersections can be read off from cone combinatorics.","The genus-zero comparison is recovered: the tropicalization of $\\mathsf{M}_{0,n}$ with the new affine structure is exactly the standard tropical space $\\mathsf{M}^{\\mathrm{trop}}_{0,n}$ with cross-ratio affine functions, so the theorem specializes to the known rational case.","For families whose moduli map lands in the good locus $\\mathsf{V}^{\\mathrm{good}}_{g,n}$, the tropicalization is a family of tropical curves and the equality holds; this gives a practical criterion for when tropical $\\psi$-class computations are trustworthy.","The extended genus-one example shows the machinery works outside the rational case: tropical $\\psi_1$ on families from admissible covers is the tropicalization of the algebraic class and agrees with operational tropicalization."],"supporting_citations":[{"why":"Defines families of tropical curves, the affine structure on tropical moduli space, and the combinatorial tropical cotangent bundle $\\mathsf{L}_i^{\\mathrm{trop}}$ that the paper compares against.","marker":"[CGM22]"},{"why":"Supplies toroidal embeddings and cone complexes with integral structures used to define tropicalization.","marker":"[KKMSD73]"},{"why":"Establishes the cone-complex tropicalization of moduli spaces of curves that the paper extends and refines.","marker":"[ACP15]"},{"why":"Provides functorial logarithmic tropicalization for log schemes, used to justify defining affine structures via log modifications.","marker":"[Uli17]"},{"why":"Supplies the notion of tropical line bundles as affine torsors and the affine-manifold framework.","marker":"[MZ08]"},{"why":"Gives the lemma that every tropical line bundle has a piecewise linear section, used in the tropical line bundle theory.","marker":"[JRS18]"},{"why":"Provides the space of tropical admissible covers and the fundamentalish cycle used in the extended example.","marker":"[CMR16]"},{"why":"Defines operational tropicalization as Minkowski weights, which the example compares against.","marker":"[Kat12]"}],"fun_headline_variants":["Tropical psi equals algebraic psi on degenerate families","Combinatorial psi classes are tropicalizations of algebraic ones","Degenerate families unify tropical and algebraic psi"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole comparison applies only to tropicalizable families, meaning the tropicalization must faithfully reflect the dual graphs of the algebraic fibers; when monodromy hides reducible fibers this fails and an étale cover is required before the theorem can be used.","fun_headline_variants_meta":{"raw":{"variants":["Tropical psi equals algebraic psi on degenerate families","Combinatorial psi classes are tropicalizations of algebraic ones","Degenerate families unify tropical and algebraic psi"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000387,"raw_usage":{"total_tokens":1977,"prompt_tokens":814,"completion_tokens":1163,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":1114}},"tokens_in":430,"tokens_out":1163,"duration_ms":11917,"temperature":1.0,"reasoning_tokens":1114,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:05:53.785574+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the tropical $\\psi_1$ class on one of the two genus-one admissible-cover families of Section 6 by evaluating the balancing condition with the affine functions pulled back from $\\mathsf{M}^{\\mathrm{trop}}_{0,5}$; the paper predicts coefficients $2/3$ and $1$ on the rays of type $\\rho_a$ and $\\rho_b$, and the pushed-forward class $12\\rho_{\\mathrm{irr}}$. A mismatch with any of these numbers would refute the comparison, while a match would confirm it in a case where the affine structure is only partially available.","supporting_citations":[],"review_version":1}