{"id":"099d4dcf-298d-4918-8921-8902eae555f4","arxiv_id":"2505.22835","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A Macaulay2 package, ToricHigherDirectImages, provides implementations for higher direct images under toric maps and Frobenius pushforwards, demonstrated on several examples.","lead":"A new Macaulay2 package computes higher direct images of line bundles under toric maps between complete toric varieties, extending beyond the earlier product-of-projective-spaces case. It also computes Frobenius pushforwards of sheaves, with examples checked against classical results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The advertised input class “surjective toric morphisms” is broader than the algorithm’s hypotheses; finite toric Frobenius maps are surjective but not toric fibrations, and the paper never states the missing condition or says the package checks it.","rationale":"The reader’s CONDITIONAL verdict is appropriate. My pass identifies the same load-bearing weakness: the missing “mild assumption” is not merely an exposition gap but a hypothesis that excludes genuine surjective toric morphisms such as toric Frobenius maps. The article moves from “toric fibration” in the technical section to “surjective toric morphisms” in the abstract and central-feature description, and no mechanism is described for checking or enforcing the stronger condition. This does not require rejecting the package, since the worked examples all appear to satisfy the fibration condition and the software is plausibly correct on its intended domain; it does require the authors to state the hypothesis precisely, add a check or warning in the package, and reconcile the version-number mismatch between the abstract and the sample session.","tokens_in":4398,"tokens_out":5967,"duration_ms":64706,"concrete_test":"In Macaulay2, load ToricHigherDirectImages and run the main higher-direct-image method on the degree-2 toric Frobenius map P^1 → P^1, obtained via phi = map(PP1, PP1, matrix{{2}}), on the structure sheaf. If the package silently returns f_*O_{P^1} = O_{P^1}, or otherwise applies the toric-fibration algorithm without rejecting the map, the central claim overreaches. Compare the output with the known Frobenius pushforward splitting from [9, Theorem 1.1]; a correct answer must be a rank-2 vector bundle, not O_{P^1}.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the package computes higher direct images of line bundles under surjective toric morphisms between complete toric varieties satisfying a mild assumption. But the article itself defines its working hypothesis as a “toric fibration”, meaning f_*O_X = O_Y, and never states the mild assumption or whether the package verifies it. This is not just a missing clarification: surjective toric morphisms do not automatically satisfy f_*O_X = O_Y. A concrete counterexample to the implicit identification is the toric Frobenius map F_p: P^1 → P^1, [x:y] ↦ [x^p:y^p], which is surjective but whose pushforward of O_{P^1} is a nontrivial sum of line bundles, not O_{P^1}; hence it is not a toric fibration. The companion paper [8] is cited for algorithms under toric fibrations, so if the package applies those algorithms to arbitrary surjective toric morphisms without checking the fibration condition, the outputs are unsupported. The article gives no proof, no external repository, and no test showing that non-fibrations are rejected or handled correctly. The abstract’s overbroad phrasing therefore directly threatens the correctness of the advertised central feature.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents the Macaulay2 package ToricHigherDirectImages, whose central feature is a method for computing (higher) direct images of line bundles under toric morphisms. The package implements algorithms from a companion paper by Roth and Zotine [8] for toric fibrations, and additionally provides a method for computing pushforwards of arbitrary sheaves under toric Frobenius maps, based on a theorem of Achinger [1]. The paper demonstrates the package on several examples, including a projective bundle over P^1, a blowup of P^2, a fibration from a smooth Fano sixfold, a Grothendieck duality verification, and a Frobenius pushforward of the cotangent sheaf on a Hirzebruch surface.","tokens_in":4643,"tokens_out":10443,"duration_ms":106792,"significance":"If the advertised functionality is correct, this package is a useful computational tool that extends the previous state of the art, which only handled maps between products of projective spaces, to arbitrary complete toric varieties satisfying a mild assumption. The examples and included assertions suggest that the implementation works for the exhibited cases, and the package's design around fans and Cox rings is natural for toric geometry. However, the central correctness claim is conditional on an unspecified 'mild assumption' and on the companion paper's algorithm; the paper does not provide an independent proof or a pinned, directly citable version of the code. The Frobenius pushforward feature is also asserted without a detailed correctness argument. Consequently, the significance is real but conditional on the missing hypotheses being stated and the implementation being reproducible.","major_comments":[{"comment":"The abstract advertises the central feature as computing higher direct images of line bundles under surjective toric morphisms, but the algorithm described in the body is only justified for toric fibrations, i.e. maps satisfying f_*O_X = O_Y. These two classes are not equivalent: the toric Frobenius map F_p: P^1 -> P^1 is surjective, yet f_*O_{P^1} is a direct sum of line bundles O(-i) for i=0,...,p-1, not O_{P^1}. The paper never states the 'mild assumption' from the abstract, nor does it say whether the package verifies the fibration condition before applying the algorithms from [8]. This is load-bearing because a user cannot know whether the method is applicable to a given ToricMap, and the package's behavior on non-fibrations is unspecified. Please state the precise hypothesis and either restrict the advertised input class to toric fibrations or explain how the package checks the condition and what it does when the condition fails.","section":"Abstract and body paragraph beginning 'Our setting is...'"},{"comment":"The claim that the package 'computes the pushforward of any module' under toric Frobenius maps is not supported by the manuscript: no algorithm or correctness statement is given beyond a reference to [1, Theorem 2] and a single example. Since the method is described as 'independent', the paper should at least sketch how the splitting from [9] is combined with a free resolution, and should state any assumptions on the toric variety (e.g., smoothness, completeness). Without this, the advertised functionality of the Frobenius method is not verifiable from the paper.","section":"Toric Frobenius Pushforwards"}],"minor_comments":[{"comment":"The abstract says the package is included in version 1.25.05 of Macaulay2, but the transcript header reads 'Macaulay2, version 1.24.05'. Please clarify which version was used for the computations and correct the mismatch.","section":"Abstract and transcript header"},{"comment":"In the derivation of f^!G = K_Z[dim Z], the composition is written 'f ◦ g' where it should be 'g ◦ f' (the structure map Z -> pt is g ∘ f). The intended argument is clear, but the typo should be fixed.","section":"Grothendieck duality example"},{"comment":"The paper does not provide a URL or stable repository for the package; adding one would facilitate reproducibility and help readers verify the claimed functionality.","section":"Throughout"},{"comment":"The notation f_*^0, f_*^1, etc. is nonstandard; please note explicitly that these denote the higher direct image functors R^0 f_*, R^1 f_*, and similarly for the package methods.","section":"Notation in examples"},{"comment":"Divisors are given as lists of integers (e.g., D = {0,0,0,0,-2,0,-2}); it would help to state explicitly that these are coefficients on the torus-invariant divisors, since this is not obvious to a reader unfamiliar with the package.","section":"Example divisor inputs"}],"recommendation":"major_revision","confidential_remarks":"The central issue is the gap between the advertised domain ('surjective toric morphisms') and the actual algorithm's hypotheses ('toric fibrations'). The 'mild assumption' must be stated explicitly, and the paper should clarify whether the package checks it. The companion paper [8], co-authored by the same author, is cited for the main algorithm; the editor may wish to consider whether the manuscript should include a more substantial description of the algorithm or at least a precise statement of its hypotheses. The package appears useful, but the manuscript as written does not yet justify the breadth of its claims. The version inconsistency is another reproducibility concern that should be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the Zotine paper. It's a Macaulay2 package highlight, not a math paper, and the honest take is that the software looks useful and the examples are convincing, but the advertised scope is broader than what the text actually claims to implement.\n\nThe genuinely new thing is the package itself: it computes higher direct images of line bundles for toric fibrations between smooth complete toric varieties, which goes beyond the earlier directImageComplex that only handled products of projective spaces. The Frobenius pushforward routine with functoriality for maps between line bundles is a separate, nice addition. The examples are well-chosen: they recover the projective bundle splitting, compute an ideal sheaf on a blowup, and verify Grothendieck duality term-by-term. The included asserts and the fact that the package ships with Macaulay2 give it real weight.\n\nThe soft spots are real but not fatal. The abstract says 'surjective toric morphisms,' but the method section immediately restricts to toric fibrations, i.e. f_*O_X = O_Y. The 'mild assumption' is never stated, and the paper doesn't say whether the package checks it or errors when it fails. That matters: a Frobenius map is surjective but not a fibration, so a user following the abstract could run the central method on inputs outside the algorithm's hypothesis and get garbage. The Frobenius section uses a different algorithm, so it's not affected, but the wording needs to be fixed. The version number mismatch (1.25.05 vs 1.24.05) is minor but should be corrected. The real proof of correctness lives in the companion paper [8]; for a software note that's acceptable, but the dependency should be stated up front and the mild assumption spelled out.\n\nI don't see a load-bearing flaw. The math is standard, the benchmarks are external, and the package appears to implement what it claims under the fibration hypothesis. The paper deserves a serious referee, mainly to check the package output and demand the language be tightened. I'd bring it to a reading group if the group cares about computational toric geometry.","headline":"Useful Macaulay2 package that genuinely extends computable higher direct images from products of projective spaces to toric fibrations, but the abstract overstates the input class and leaves the key 'mild assumption' unspecified.","tokens_in":5136,"tokens_out":2910,"would_cite":true,"duration_ms":27107,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14M25","14Q15","13-04"],"pacs":[],"model":"deepseek-v4-flash","headline":"This article presents a computer-algebra package that computes higher direct images of line bundles under surjective toric morphisms, replacing a method that only worked for products of projective spaces.","keywords":["higher direct images","toric varieties","toric fibrations","Frobenius pushforwards","line bundles","Čech complexes","computer algebra package","coherent duality"],"falsifier":"Take a toric fibration with a hand-computable answer, such as a projective bundle over a complete toric threefold pushed forward with the structure sheaf, and compare the package's $R^i f_* \\mathcal{O}_X$ to the classical projective-bundle formula; any mismatch in rank, twist, or filtration would refute the claimed generality. A second test would be to run the method on a map violating the unspecified mild assumption and check whether the output contradicts known local-cohomology vanishings.","tokens_in":4192,"feed_emoji":"📦","tokens_out":8507,"duration_ms":95774,"temperature":0.7,"pith_summary":"This article presents a computer-algebra package that computes higher direct images of line bundles under surjective toric morphisms. Higher direct images generalize sheaf cohomology to the relative setting, measuring sections that fail to push forward along a map, and they are central invariants in algebraic geometry. A previous implementation could only handle maps between products of projective spaces; the new package works for maps between any complete toric varieties satisfying a mild assumption. The article also adds an independent algorithm for Frobenius pushforwards of arbitrary sheaves, obtained by resolving sheaves by line bundles and pushing the resolution forward. Worked examples reproduce classical splitting behavior on projective bundles, recover ideal sheaves of blown-up points, and verify coherent duality term by term.","feed_headline":"New package computes higher direct images on toric maps","feed_subtitle":"Previously limited to products of projective spaces; now any complete toric fibration under a mild assumption is in reach.","key_machinery":"The load-bearing machinery is the reduced Čech complex of the companion paper [8]—a trimmed Čech complex that computes the same higher direct images without redundant sheaves—together with the character decomposition of $R^i f_* L$ into eigensheaves indexed by characters of the kernel torus $T_K$. For Frobenius pushforwards, the machinery is the line-bundle splitting theorem [9], implemented through the combinatorial description of [1], extended by functoriality and free resolutions so that arbitrary coherent sheaves can be pushed forward.","core_discovery":"The central claim is that the ToricHigherDirectImages package faithfully implements the companion algorithm [8], which computes $R^i f_* L$ for a line bundle $L$ on a smooth complete toric variety $X$ under a surjective toric morphism $f : X \\to Y$ to another smooth complete toric variety. The computation is organized by the kernel torus $T_K = \\ker f|_{T_X}$: each $R^i f_* L$ splits into eigensheaves indexed by characters of $T_K$, and the package's computeEigencharacters method returns the divisors on $X$ and $Y$ needed to build those summands. The package also implements a separate Frobenius pushforward routine: for the $p$-th toric Frobenius map, the splitting theorem [9] describes how the pushforward of a line bundle splits into line bundles, and the algorithm from [1] makes this explicit; functoriality lets the user push forward any sheaf by resolving it by line bundles and pushing forward the complex.","pith_inferences":["If the companion algorithm's coverage extends beyond the smooth case, the same character-indexed decomposition should work for singular complete toric varieties; pushing a few known singular examples through the package would be a direct test.","The Frobenius routine is characteristic-independent by construction, so it could be used to probe phenomena special to positive characteristic, such as p-torsion in pushforwards, without needing a finite-field implementation.","A full complex-level implementation of shriek functors would turn the term-by-term duality checks into a derived-category tool; the examples here suggest the package already computes all the pieces."],"forward_implications":["Any complete toric fibration satisfying the mild assumption becomes computationally accessible, not just maps between products of projective spaces.","The character decomposition output by computeEigencharacters makes the eigensheaf structure of each higher direct image explicit, which is exactly the data needed for further cohomological calculations.","Frobenius pushforwards of arbitrary coherent sheaves can now be computed, giving a concrete handle on objects of recent interest in birational geometry and derived categories.","Coherent duality can be checked term by term for line bundles whose higher direct images live in a single degree, as demonstrated on a Fano sixfold fibration over projective four-space.","The package is distributed with the computer algebra system's version 1.25.05, so the algorithms are available to other users directly."],"supporting_citations":[{"why":"Supplies the reduced Čech complex algorithm for higher direct images that the package implements.","marker":"[8]"},{"why":"Provides the previous state-of-the-art directImageComplex method, which only worked for maps between products of projective spaces.","marker":"[5]"},{"why":"Gives the theorem and combinatorial description on which the Frobenius pushforward algorithm is based.","marker":"[1]"},{"why":"States the splitting theorem that describes the summands of Frobenius pushforwards of line bundles.","marker":"[9]"},{"why":"Supplies the standard definitions of pushforward and higher direct image and the classical projective-bundle behavior used in examples.","marker":"[7]"},{"why":"Provides the coherent duality statement that the package verifies term by term.","marker":"[6]"},{"why":"Establishes the fan-to-linear-map correspondence used to encode toric varieties and toric morphisms in the package.","marker":"[4]"}],"fun_headline_variants":["Toric direct images: now any fibration, not just products","Macaulay2 computes all R^i f_* for toric fibrations","Beyond products: toric higher direct images in Macaulay2","ToricHigherDirectImages: computes R^i f_* on toric fibrations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole package rests on the companion algorithm [8] being valid for every surjective toric morphism between complete toric varieties that satisfies the 'mild assumption'—an assumption this article never actually states.","fun_headline_variants_meta":{"raw":{"variants":["Toric direct images: now any fibration, not just products","Macaulay2 computes all R^i f_* for toric fibrations","Beyond products: toric higher direct images in Macaulay2","ToricHigherDirectImages: computes R^i f_* on toric fibrations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001254,"raw_usage":{"total_tokens":5053,"prompt_tokens":772,"completion_tokens":4281,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":388,"completion_tokens_details":{"reasoning_tokens":4198}},"tokens_in":388,"tokens_out":4281,"duration_ms":29900,"temperature":1.0,"reasoning_tokens":4198,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T12:58:53.802321+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a toric fibration with a hand-computable answer, such as a projective bundle over a complete toric threefold pushed forward with the structure sheaf, and compare the package's $R^i f_* \\mathcal{O}_X$ to the classical projective-bundle formula; any mismatch in rank, twist, or filtration would refute the claimed generality. A second test would be to run the method on a map violating the unspecified mild assumption and check whether the output contradicts known local-cohomology vanishings.","supporting_citations":[{"cited_title":"Reduced \\v{C}ech complexes and computing higher direct images under toric maps","cited_arxiv_id":"2504.12903","evidence_quote":"Supplies the reduced Čech complex algorithm for higher direct images that the package implements."},{"cited_title":"Relative Beilinson monad and direct image for families of coherent sheaves","cited_arxiv_id":null,"evidence_quote":"Provides the previous state-of-the-art directImageComplex method, which only worked for maps between products of projective spaces."},{"cited_title":"A characterization of toric varieties in characteristic p","cited_arxiv_id":null,"evidence_quote":"Gives the theorem and combinatorial description on which the Frobenius pushforward algorithm is based."},{"cited_title":"Frobenius direct images of line bundles on toric varieties","cited_arxiv_id":null,"evidence_quote":"States the splitting theorem that describes the summands of Frobenius pushforwards of line bundles."},{"cited_title":"Algebraic Geometry","cited_arxiv_id":null,"evidence_quote":"Supplies the standard definitions of pushforward and higher direct image and the classical projective-bundle behavior used in examples."},{"cited_title":"Residues and duality, volume No","cited_arxiv_id":null,"evidence_quote":"Provides the coherent duality statement that the package verifies term by term."},{"cited_title":"Cox, John B","cited_arxiv_id":null,"evidence_quote":"Establishes the fan-to-linear-map correspondence used to encode toric varieties and toric morphisms in the package."}],"review_version":1}