{"id":"4849e9e4-eb0e-4783-a100-43ae42145e4f","arxiv_id":"2506.21268","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local dimension of a tropical linear system is bounded below by its Baker-Norine rank, and the realizable canonical divisors form a tropically convex, definable, closed polyhedral complex.","lead":"This mathematics thesis proves structural results about linear systems on metric graphs, which are geometric models of what happens when algebraic curves degenerate. It shows that the local dimension of a tropical linear system is always at least its rank, and that the set of realizable canonical divisors forms a well-behaved geometric complex.","discovery_kind":"extension","skeptic_critique":null,"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies linear systems on metric graphs, establishes a lower bound on the local dimension of a tropical linear system in terms of the Baker–Norine rank, analyzes the structure of the canonical linear system, and investigates the realizability locus of canonical divisors. The main results are: (i) the dimension of every maximal face of a finitely generated tropical linear system is at least its rank (Cor. 2.95.2); (ii) the realizability locus Real(|K|) of the canonical linear system is tropically convex, definable, and closed, hence an abstract polyhedral complex (Props. 3.21–3.23); (iii) under a condition excluding two disjoint horizontal cycles, every canonical divisor is realizable, and in particular some (g−1)-dimensional maximal cell lies in the realizability locus (Prop. 3.27, Cor. 3.29.1); and (iv) tropicalizations of linear series of rank r on algebraic curves are equi-dimensional polyhedral complexes of dimension r (Cor. 3.34.2). The exposition is detailed and the paper includes computer-assisted examples and a GitHub implementation.","tokens_in":37137,"tokens_out":10954,"duration_ms":128850,"significance":"If the results are correct, the paper contributes a useful structural link between combinatorial rank and geometric dimension in tropical linear series, and it gives a concrete polyhedral description of a nontrivial realizability locus. The main strengths are the clean induction in Prop. 2.78, the careful treatment of generic divisors and non-splitting divisors, and the use of the external benchmarks (BN07, MUW17, JP22, FJP23) without fitted parameters or circular reasoning. The paper is honest about the totally degenerate setting (h ≡ 0) in which the realizability-locus results are proved. However, the proof of the horizontal-edge case in Prop. 3.27 and the closedness argument in Prop. 3.23 contain gaps, and the passage from tropical independence of divisors to the quoted upper bound of [JP22] needs clarification; these issues affect load-bearing claims, so the paper requires revision before its central conclusions can be considered established.","major_comments":[{"comment":"The horizontal-edge part of the proof is incomplete. After reducing to the case where C \\ e is disconnected and choosing C_1, C_2, the text considers a leaf x of C and constructs a monotone path from x to a local maximum containing a horizontal cycle, but it never produces the promised simple path from v_i to that cycle, nor does it show that the cycles obtained from C_1 and C_2 can be connected through e into a single simple cycle lying above e. The sentence 'like before, this would prove that e is contained in a simple cycle that lies above it' asserts the desired conclusion. Since Proposition 3.27 is used in Corollary 3.29.1 to place a whole (g−1)-dimensional cell inside the realizability locus, this missing step is load-bearing and should be supplied.","section":"§3.5, Prop. 3.27"},{"comment":"In the closedness proof for horizontal edges, the condition '∥f−fn∥ ≤ l(e) < 2' is not the correct hypothesis: edge lengths are arbitrary, and the relevant condition is that the uniform distance is smaller than l(e), not that l(e) < 2. More importantly, showing that f_n has a horizontal section on a subsegment of e does not imply that e itself is a horizontal edge in the model to which Theorem 3.19 is applied, nor does it guarantee that the realizable cycle γ_n contains e. The limit argument needs a cycle in Γ containing e and satisfying f(γ) ≥ f(e); as written this step does not go through. This gap affects the proof that Real(|K|) is closed and hence Corollary 3.23.1.","section":"§3.4, Prop. 3.23"},{"comment":"There is a mismatch between the objects in the two bounds being combined. By Definition 2.86, d is a tropically convex subset of the projectivized linear system |D|, not a module; the associated module is R(d,D). Prop. 3.34 is phrased as 'Let d ⊆ |D| be a finitely generated submodule' and speaks of functions of d, while Definition 3.32 defines tropical dependence for subsets of |D| (divisors). Prop. 3.33 gives tropical dependence for points of trop(d_X), not for functions in its cone. Cor. 3.34.1 therefore does not follow as written. The authors should state the module-level version of [JP22, Cor. 4.7], verify that Prop. 3.33 applies to the same object, and then pass to the projectivization. Without this, the central equi-dimensionality claim Cor. 3.34.2 is not established.","section":"§3.6, Props. 3.33–3.34 and Cor. 3.34.1"}],"minor_comments":[{"comment":"The paragraph 'In section 4 we make the links between the worlds of tropical and algebraic geometry' actually describes Section 3; Section 4 is about discrete representations. The section numbers should be corrected.","section":"§1, Structure of the thesis"},{"comment":"In the definition of the length of a path, the summand is written as d(γ(x_{i−1}), γ(y_i)); the second argument should presumably be γ(x_i).","section":"§2.1, Definition 2.1"},{"comment":"The abstract says the work 'provides a characterization of realizable canonical divisors', but the full characterization is quoted from [MUW17] and the new results are structural and sufficient conditions in the totally degenerate case. The abstract and the introduction should state this scope explicitly.","section":"Abstract and §3.4"},{"comment":"The restriction to h ≡ 0 is justified by a density statement in the moduli space, but the paper should make clear in the main theorems that the polyhedral-complex and tropical-convexity statements about Real(|K|) are proved only in this totally degenerate setting and are not claimed for arbitrary vertex-weighted metric graphs.","section":"§3.4, beginning"},{"comment":"In the sentence 'the set of a_i such that max(a_i + φ_i) ≥ 0 on a fixed edge', the maximum should be over i; as written the expression is ambiguous. Also the notation PΩMtrop_g should be defined at first use.","section":"§3.4, Prop. 3.22"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a carefully written master's thesis with several genuine lemmas and useful examples, and the gaps identified above appear fixable rather than fatal. The main concern is that two of the paper's headline conclusions—the closedness of the realizability locus and the equi-dimensionality of tropicalizations of linear series—currently rest on arguments that are either incomplete or stated for the wrong object. I would advise a major revision that supplies the missing proof steps in §3.5 and §3.4 and clarifies the module-versus-projectivization issue in §3.6. The GitHub code and the explicit counterexamples are a genuine strength and should be kept."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this is a competent, carefully written Master's thesis. The genuinely new content is the dimension lower bound for tropical submodules (Prop 2.78 / Cor 2.95.1) and the structural analysis of the realizable canonical locus (Props 3.21–3.27, Cor 3.29.1). Those results are not in MUW17, JP22, or FJP23. The lower-bound proof is clean, and the definability/closedness arguments for Real(|K|) are sound.\n\nWhat the paper does well: it gives full detail, writes out the polyhedral complex machinery, and ships code that lets you test the ideas. The specialization chapter is a fair survey, and the equi-dimensionality corollary (3.34.2) follows from the lower bound plus the JP22/FJP23 theorems. The citation pattern is normal; no self-citation loops, no fitted parameters.\n\nThe soft spots are real but modest. Prop 3.27's horizontal-edge leaf case is sketched; the concluding step should be written out. Prop 2.85 asserts that for a generic D near the constructed S we necessarily have g(Γ \\ supp D_E) ≤ 1; that step deserves a line of justification. The bigger caveat is structural: the realizability-locus results are proved only for h ≡ 0 and take the MUW17 criterion as a black box. That restriction is acknowledged in the body (Section 3.4) and is reasonable for a thesis, but it means the structural claims about Real(|K|) are about totally degenerate graphs only. The abstract overstates slightly: it says the paper provides a characterization of realizable canonical divisors, when the characterization is MUW17's and this paper gives a cleaner criterion plus sufficient conditions. That should be fixed in revision.\n\nI agree with the reader's conditional verdict. The main theorems are plausible, the proofs are mostly detailed, and the gaps are patchable. Anyone working on tropical linear series or the specialization bridge will get value here. It deserves a serious referee.\n\nRecommendation: send it to peer review. If I were handling it, I'd ask the author to expand the two proof passages and reword the abstract. The central claims look correct.","headline":"A solid Master's thesis with real new results (dimension lower bound, structure of the canonical realizability locus) and two small proof gaps that are patchable; the abstract oversells the characterization slightly.","tokens_in":37724,"tokens_out":2003,"would_cite":true,"duration_ms":22777,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14T10","14H51"],"pacs":[],"model":"deepseek-v4-flash","headline":"Tropical linear systems have dimension at least their rank, and tropicalizations of algebraic linear series have dimension exactly their rank.","keywords":["tropical geometry","metric graphs","linear systems","Baker–Norine rank","tropical modules","canonical divisors","realizability","abstract polyhedral complexes"],"falsifier":"Find a finitely generated tropical linear system d with a maximal face of dimension strictly less than r(d); the dense non-splitting induction in Proposition 2.95 would have to break at that face. For the realizability statements, test the quoted realizability criterion on an enhanced level graph with a vertex weight h>0: if the criterion misclassifies a canonical divisor there, the structural theorems about Real(|K|) do not transfer to weighted graphs.","tokens_in":36983,"feed_emoji":"📐","tokens_out":7137,"duration_ms":76443,"temperature":0.7,"pith_summary":"This paper proves a rank–dimension theorem for linear systems on metric graphs: the Baker–Norine rank, which counts how many chips can be removed while keeping the system non-empty, is always a lower bound for the local dimension of a finitely generated tropical linear system, and for systems obtained by tropicalizing linear series on algebraic curves the two numbers are equal, making the systems equi-dimensional polyhedral complexes of that dimension. The paper also attacks the realizability problem for canonical divisors: using the quoted realizability criterion from [MUW17], it characterizes realizable divisors in the canonical linear system in terms of simple cycles lying above inconvenient vertices and horizontal edges, and shows from this that the realizable locus is tropically convex, closed, definable, and therefore an abstract polyhedral complex. It further shows that every metric graph without two disjoint horizontal cycles has all effective canonical divisors realizable, while graphs with disjoint cycles always contain non-realizable canonical divisors, and that the realizable locus always contains a maximal cell of dimension g-1. A final section translates the theory to finite graphs with unit edge lengths and develops algorithms for enumerating divisors, finding extremals, and checking realizability.","feed_headline":"Tropicalized linear series have dimension equal to rank","feed_subtitle":"New lower bound ties Baker–Norine rank to cell dimension; canonical divisors are realizable exactly when cycles cover their problem…","key_machinery":"The engine of the lower bound is the tropical module R(D) = {f ∈ Rat(Γ) : D+div(f) ≥ 0}, whose tropical projectivization is the complete linear system |D|; a tropical linear system d is a tropically convex subset of |D|, and its rank r(d) is defined by requiring d(-E) non-empty for every effective divisor E of degree d. The proof builds tangent vectors from the capped functions ft = f ⊕ (sup f - t), which act as chip-firing moves, and uses 'non-splitting' divisors, for which no function in the system moves a chip off the boundary of its maximum locus; these are dense and their tangent spaces have independent directions for each removed chip. For realizability, the load-bearing criterion, quoted from [MUW17], says an effective canonical divisor K+div(f) on a metric graph is realizable exactly when every inconvenient vertex and every horizontal edge is contained in a simple cycle that lies above it; an inconvenient vertex is one of weight zero whose outgoing slopes are all nonzero and where some negative slope exceeds the sum of the positive slopes.","core_discovery":"The central claim is that local tropical dimension and Baker–Norine rank are locked together for well-behaved tropical linear systems. The paper proves that if a tropical linear system d is finitely generated, its maximal cells all have dimension at least r(d); the proof works by showing that a divisor that 'does not split' has a tangent space of dimension at least r(d), and such divisors are dense. When d is the tropicalization of a rank r linear series on an algebraic curve, the upper bound from tropical independence forces every maximal cell to have dimension exactly r, so the tropicalization is equi-dimensional of dimension r. On the realizability side, the paper proves that a canonical divisor K+div(f) is realizable precisely when every inconvenient vertex and every horizontal edge is contained in a simple cycle on which f is at least as large, and derives from this that the realizable locus Real(|K|) is a tropically convex, closed, definable subset of |K|, hence an abstract polyhedral complex; it always contains a maximal cell of dimension g-1.","pith_inferences":["The upper bound from [JP22] requires a tropical-independence condition, so equality for tropicalized series suggests that realizable systems are exactly those where independence and rank coincide; testing this on the author's code could reveal whether the independence condition is also necessary.","The paper leaves open whether Real(|K|) is finitely generated; a positive answer would turn the realizability characterization into an explicit finite description via extremals of the tropical module.","The restriction to totally degenerate graphs (vertex weight h=0) is the main barrier: canonical divisors are usually defined with vertex weights, and a weighted counterexample to the quoted realizability criterion would force revisiting the density step before applying these results to stable curves.","The discrete algorithms on unit-length models suggest that rank and realizability can be checked exhaustively on finite graphs; scaling to finer subdivisions should give computable approximations for arbitrary rational divisors on metric graphs."],"forward_implications":["Finitely generated tropical linear systems are abstract polyhedral complexes whose maximal faces all have dimension at least the Baker–Norine rank.","Tropicalizations of rank r linear series on algebraic curves are equi-dimensional polyhedral complexes of dimension r, so for realizable systems rank equals dimension.","A canonical divisor on a metric graph with no two disjoint horizontal cycles is always realizable; conversely, two disjoint cycles force the existence of non-realizable canonical divisors.","The realizable locus in a canonical linear system is tropically convex and an abstract polyhedral complex, and it contains a maximal cell of dimension g-1.","The lower bound can fail for arbitrary complete linear systems: the dumbbell graph has cells of dimension strictly larger than the rank, so the rank–dimension equality is special to tropicalized systems."],"supporting_citations":[{"why":"Supplies the realizability criterion for canonical divisors and the density of the h=0 locus; it is the black box on which all realizability-locus results rest.","marker":"[MUW17]"},{"why":"Gives the upper bound on dimension via tropical independence and establishes that finitely generated tropical linear series are closed definable subsets.","marker":"[JP22]"},{"why":"Shows tropicalizations of algebraic linear series are finitely generated of rank at least r and satisfy tropical dependence, used for the equi-dimensionality corollary.","marker":"[FJP23]"},{"why":"Defines the Baker–Norine rank and proves tropical Riemann–Roch; the rank is the quantity the dimension lower bound is compared against.","marker":"[BN07]"},{"why":"Provides the specialization lemma and the specialization map from curves to metric graphs, the bridge used throughout the realizability discussion.","marker":"[Bak07]"},{"why":"Supplies finite generation of R(D), extremals, and the decomposition of rational functions into chip-firing moves used in the lower-bound proof.","marker":"[HMY09]"},{"why":"Gives the abstract polyhedral complex structure of complete linear systems and the reduced-divisor machinery used in the discrete section.","marker":"[Ami12]"}],"fun_headline_variants":["Tropical linear system dimension equals Baker–Norine rank","Realizable canonical divisors characterized by cycle condition","Tropical linear series: dimension equals rank","Baker–Norine rank bounds tropical submodule dimension","Equi-dimensional tropicalization of algebraic linear series"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The realizability-locus results are proved only for totally degenerate graphs, where every vertex has genus weight h=0, and they rely on the quoted Moeller–Ulirsch–Werner realizability criterion as a black box; if that criterion needs modification for vertex-weighted graphs, or if the density statement used to pass from h=0 to arbitrary weights fails, the polyhedral structure of Real(|K|) in the weighted setting is not established.","fun_headline_variants_meta":{"raw":{"variants":["Tropical linear system dimension equals Baker–Norine rank","Realizable canonical divisors characterized by cycle condition","Tropical linear series: dimension equals rank","Baker–Norine rank bounds tropical submodule dimension","Equi-dimensional tropicalization of algebraic linear series"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000203,"raw_usage":{"total_tokens":1314,"prompt_tokens":800,"completion_tokens":514,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":416,"completion_tokens_details":{"reasoning_tokens":442}},"tokens_in":416,"tokens_out":514,"duration_ms":5592,"temperature":1.0,"reasoning_tokens":442,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:30:17.675437+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a finitely generated tropical linear system d with a maximal face of dimension strictly less than r(d); the dense non-splitting induction in Proposition 2.95 would have to break at that face. For the realizability statements, test the quoted realizability criterion on an enhanced level graph with a vertex weight h>0: if the criterion misclassifies a canonical divisor there, the structural theorems about Real(|K|) do not transfer to weighted graphs.","supporting_citations":[],"review_version":1}