{"id":"4a0563c8-bd10-488b-bb17-33b4a196ae70","arxiv_id":"2501.06960","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every nef divisor on a smooth projective toric variety, the paper constructs a cellular resolution of a new ceiling truncation ideal and shows it agrees with the Hanlon-Hicks-Lazarev Fourier-Mukai transform.","lead":"This paper builds explicit short free resolutions for truncated coordinate rings on any smooth projective toric variety, and proves they match a construction that came from symplectic geometry. It also finds examples where that construction has unexpected nonzero homology, which is new for the commutative algebra analogue.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1's isomorphism depends on a globally M-invariant incidence function on C that is never constructed; the proof matches terms via Lemma 5.2 but leaves boundary-map compatibility with multiple preimages unchecked.","rationale":"The paper is serious and well-motivated: Theorem 2.5 has a clean convexity argument, and the fully worked F2 example is credible computational evidence. I read Theorem 5.1 as the central bridge identifying the elementary cellular complex with the HHL Fourier-Mukai transform, so its proof is the most load-bearing part of the paper. The term-level bijection in Lemma 5.2 is plausible, but the boundary-map identification is where the argument is thinnest. The paper itself flags the obstruction: quotients of the infinite cell complex can identify distinct preimages, and global lifting of boundary incidences is impossible in general. Theorem 5.1 requires a global M-invariant incidence function whose restriction to D and descent to E produce identical boundary maps; the proof verifies only a single local incidence and asserts the rest. This is an internal completeness gap rather than a disagreement with consensus, and it is exactly the reader's weakest assumption, so I agree with the reader's CONDITIONAL verdict. A concrete entry-by-entry comparison in the F2 example, using an explicitly chosen eps, would either expose a mismatch or provide the missing evidence for the gluing claim.","tokens_in":12482,"tokens_out":19384,"duration_ms":206446,"concrete_test":"For the Hirzebruch surface F2 example of Section 4 with d=(1,1), fix one explicit M-invariant incidence function eps on the cubical subdivision C (e.g., by orienting the standard coordinate hyperplane arrangement), compute the boundary matrices of F(d)bullet and Phi(S(d)) from Constructions 2.2 and 3.1 using this same eps, and compare them entry-by-entry under the bijection of Lemma 5.2. Any sign or multiplicity mismatch, especially for the horizontal edge with two preimages in E, would show Theorem 5.1 fails; a full match would supply the missing gluing evidence for that case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the map-level isomorphism in Theorem 5.1. Lemma 5.2 supplies a bijection of summands, but the differentials are only matched locally. The paper states in Section 3, just before (3.1), that in general no global set of representatives can be chosen so that every boundary incidence in E lifts to a boundary incidence in C, and Section 4 gives an explicit instance: the horizontal edge in E has its two ends identified to the same vertex, so there are two local contributions to the boundary map. For the isomorphism F(d)bullet = Phi(S(d)) to hold, a single incidence function eps on C must be invariant under translation by M and must induce the same boundary maps on the slice D and on the quotient E, including all preimages. The proof of Theorem 5.1 says 'using incidence functions from the same eps on C' and then checks one incidence pair; it does not prove that the local choices glue to a global M-invariant eps, nor that no extra terms or sign mismatches arise from cells with multiple preimages. If such an eps cannot be chosen, the isomorphism with the HHL resolution, and with it Corollaries 5.3 and 6.5, is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a notion of ceiling truncation for the total coordinate ring S of a smooth projective toric variety X, and constructs, for each d in the nef cone, a finite cellular complex F_• supported on the polytope of sections of O(d). Theorem 2.5 asserts that F_• is a free resolution of the ceiling truncation truncd(S), with length at most dim X. In Section 5 the author claims that after twisting by d, F(d)_• is isomorphic to the Fourier-Mukai transform of O(d) with respect to the Hanlon-Hicks-Lazarev resolution of the diagonal. Section 6 defines a truncation functor for arbitrary modules and uses it, via a spectral sequence argument, to compare homology of the transform with homology of the truncated resolution; Example 6.7 reports nontrivial homology for all positive multiples of an ample class. The paper also contains a fully worked example for the Hirzebruch surface F2.","tokens_in":12795,"tokens_out":4866,"duration_ms":48487,"significance":"If the main comparison theorem is fully established, the paper gives a valuable explicit and computable analogue of Eagon-Northcott complexes for nef truncations on all smooth projective toric varieties, and it connects commutative algebra with the symplectically motivated short virtual resolutions of Hanlon, Hicks, and Lazarev. The convexity proof of Theorem 2.5 is clean and appears correct, and the use of Macaulay2 in Section 4 is a useful computational check. The claimed nontrivial homology in Section 6 would be a genuinely new phenomenon if supported by a complete argument. However, the map-level comparison in Theorem 5.1 and the degeneration claim in Corollary 6.5 are not fully proved as written, and Example 6.7 does not contain the promised hand-checkable proof for all multiples. These issues are load-bearing for the paper's central claims and require revision.","major_comments":[{"comment":"The proof of Theorem 5.1 asserts an isomorphism of complexes F(d)_• ≅ Φ(S(d)) using 'incidence functions from the same ε on C', but no such globally M-invariant incidence function is constructed. Lemma 5.2 establishes only a one-to-one correspondence between open cells of D and monomials; it does not compare boundary maps when cells of E have multiple preimages in C. The text immediately before (3.1) explicitly says that no global set of representatives can be chosen so that every boundary incidence in E lifts to a boundary incidence in C, and Section 4 displays a horizontal edge in E whose two ends are identified to the same vertex. For a map-level isomorphism one must prove that a single ε on C is invariant under translation by M and induces the same cellular boundary maps on D and on E, including all preimages and sign choices. As written, only one incidence pair is checked, so the isomorphism, and with it Corollaries 5.3 and 6.5, is not fully established. Please either construct ε explicitly, prove its existence by a global orientation argument, or revise the statement to a weaker comparison that does not require a global incidence function.","section":"§5, proof of Theorem 5.1; §3, (3.1)"},{"comment":"Example 6.7 states that 'homology can be exhibited by hand for all m' and identifies generators arising from the term R(1,-1,1,-1) in G1, but no calculation is given. This is load-bearing: the claimed persistence of homology for all multiples of an ample divisor is the paper's main new phenomenon, stated in the final paragraph of Section 6. A Macaulay2 check for finitely many m does not establish the infinite family. Please include the hand calculation or provide a complete supplementary argument proving the stated vanishing failure for every m.","section":"§6, Example 6.7"},{"comment":"The proof of Corollary 6.5 contains the sentence 'In the spectral sequence of the double complex all maps are zero starting at the second page for both directions' with no justification. The degeneration of this spectral sequence is exactly what identifies the homology of truncd(K_•)(d) with that of Φ(Q(d)); without a proof or a reference to a proved lemma, the corollary is not established. Please supply the computation of the second page or state the missing hypothesis that makes the degeneration immediate.","section":"§6, proof of Corollary 6.5"}],"minor_comments":[{"comment":"There is a typographical error: 'analagous' should be 'analogous' in the sentence about pictures from symplectic geometry.","section":"Introduction, §4"},{"comment":"Figure 5 is difficult to parse: the caption lists 'edges in D F1 G1 Φ(S(d))1 polytope in S' without indicating which columns correspond to which complex. Please restructure the figure or caption so the correspondence of summands is explicit.","section":"Figure 5 and surrounding text"},{"comment":"Remark 6.4 says 'the author believes that a general definition is possible'; this speculative statement should be removed or replaced by a precise conjecture, since it is not used in the paper.","section":"Remark 6.4"},{"comment":"The incidence function on D is said to be 'chosen by Macaulay2'; for reproducibility, please provide the code or explicitly list the orientations of all cells in Figures 3 and 4.","section":"Section 4"},{"comment":"Reference [And23] is listed as 'Pre-published' with a 2024 arXiv number; please update its publication or arXiv status.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the global incidence function in Theorem 5.1. If the author can supply a proof of existence of an M-invariant ε with the required compatibility, or suitably weaken the statement, the paper is likely publishable. The proof of Corollary 6.5 and the claimed all-m computation in Example 6.7 also need to be completed. The convexity part and the overall framework are sound and useful."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Those two theorems you asked about: Theorem 2.5 is a clean, self-contained result and I think it's correct. The label-by-ceiling trick and the convexity argument for contractibility are exactly the right tools. That part is solid and genuinely new for non-unimodular varieties. Theorem 5.1 is the more important claim—the resolution agrees with the HHL Fourier–Mukai transform—but the proof has a load-bearing gap. The paper needs a global M-invariant incidence function on the infinite cell complex C that restricts correctly to both the polytope D and the quotient torus E. Lemma 5.2 gives a bijection of summands, and the local map check works, but the boundary maps are only matched locally. Section 3 itself points out that you can't always pick global representatives, and Section 4's example shows a cell in E whose boundary identifications create multiple preimages. So the differentials might not glue the way the proof assumes. The stress-test note has this right: the existence of such an epsilon is assumed, not proved. I'd like to see that step spelled out or a counterexample. Maybe it's fixable—the cellular geometry is amenable—but as written the isomorphism is not fully established.\n\nThat said, the rest is in good shape. The ceiling truncation functor in Section 6 is a reasonable extension, the corollaries follow formally once Theorem 5.1 is granted, and the homology example is striking. I do wish Example 6.7 included the promised hand proof for all multiples of d; right now it's a Macaulay2 computation plus a note that it can be done by hand. A referee should ask for that.\n\nCitations are fair and the paper gives proper credit to HHL, BPS, and the others. No fitting, no circularity. The author is clearly thinking carefully about the geometry.\n\nWho is this for? People working on virtual resolutions, multigraded syzygies, and toric constructions. It's a useful paper even with the gap, because the explicit construction and the ceiling truncation definition will probably stand. I'd send it to a serious referee, with the expectation that the incidence function issue gets resolved or explained.","headline":"A valuable explicit construction of ceiling truncation resolutions on toric varieties, with a genuine gap in the isomorphism theorem that needs patching before the main comparison is airtight.","tokens_in":13251,"tokens_out":1983,"would_cite":true,"duration_ms":18234,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["13D02","14M25"],"pacs":[],"model":"deepseek-v4-flash","headline":"For every nef divisor on a smooth projective toric variety, the paper constructs an explicit cellular free resolution of a ceiling truncation ideal, of length at most the dimension, and shows it matches a symplectic-geometry construction.","keywords":["toric varieties","cellular resolutions","ceiling truncation","short virtual resolutions","Fourier-Mukai transforms","syzygies","total coordinate rings","Eagon-Northcott complexes"],"falsifier":"Choose a smooth toric variety where the quotient cell complex $E$ identifies multiple preimages of a cell (as in the worked example), fix two incidence functions on $C$ that are invariant under translation by $M$, compute the boundary maps of $F(d)_\\bullet$ for both, and compare their homologies; if the homologies differ, or if $\\partial^2\\neq 0$ for either one, the central resolution and isomorphism claims fail.","tokens_in":12318,"feed_emoji":"📐","tokens_out":8287,"duration_ms":76195,"temperature":0.7,"pith_summary":"This paper gives every nef divisor on a smooth projective toric variety an explicit free resolution of a naturally defined monomial ideal, with length no more than the dimension of the variety. The ideal, called the ceiling truncation of the total coordinate ring, is generated by the ceiling labels of a finite cell complex built from the polytope of global sections. The complex is shown to be a cellular resolution, and after twisting by the divisor it reproduces the Fourier–Mukai transform of $\\mathcal O(d)$ along a known resolution of the diagonal. The same combinatorial truncation extends to arbitrary modules and computes the homology of the transformed modules; the author exhibits a smooth toric variety where this homology is nonzero for every positive multiple of an ample divisor, so exactness can fail at the level of modules. If correct, the construction is the missing toric analogue of the Eagon–Northcott resolution of powers of the maximal ideal in projective space.","feed_headline":"Every nef divisor on a toric variety gets an explicit short resolution","feed_subtitle":"The celular complex matches the symplectic Fourier–Mukai construction and exposes new module-level homology.","key_machinery":"The engine is the alcove cell structure obtained by subdividing $\\mathbb R^r$ along all integral translates of the coordinate hyperplanes, then restricting to the affine plane $\\alpha + M\\otimes\\mathbb R$ and to the positive quadrant, producing the finite labeled cell complex $D$. Each cell is labeled by rounding fractional exponents upward to the nearest integer, and a lemma shows that every cell label is the least common multiple of its boundary labels; the ideal generated by these labels is the ceiling truncation $\\operatorname{trunc}_d(S)$. The complex $F_\\bullet$ is the cellular complex of $D$ with respect to an incidence function. The isomorphism with the Fourier–Mukai transform rests on a bijection between open cells of $D$ lying over a torus cell and monomials of degree $d+\\deg x^{\\lfloor\\gamma\\rfloor}$, which unfolds the torus cell complex into the polytope of sections.","core_discovery":"The central claim is that for every nef divisor $d$ on a smooth projective toric variety $X$, the monomial ideal $\\operatorname{trunc}_d(S)$ generated by the ceiling labels of an explicitly defined labeled cell complex $D$ inside the polytope of sections of $\\mathcal O(d)$ has a cellular free resolution $F_\\bullet$ of length at most $\\dim X$. After twisting by $d$, this same complex sheafifies to the Fourier–Mukai transform of $\\mathcal O(d)$ with respect to a recently constructed resolution of the diagonal, giving an explicit algebraic model of that transform for structure sheaves of nef divisors. The construction also yields a truncation functor on modules: for a module $Q$ satisfying a nefness condition, the homology of the ceiling-truncated free resolution of $Q$ computes the homology of the Fourier–Mukai transform of $Q(d)$. A notable consequence is that this homology need not vanish: on an explicit smooth toric variety, the transform of $Q(md)$ has nontrivial homology for all sufficiently large multiples $m$, so exactness can fail at the module level even when the sheafified complex is exact.","pith_inferences":["The construction suggests a practical algorithmic route to short virtual resolutions of line bundles on any smooth projective toric variety, avoiding spectral sequences and mirror functors, and it seems likely to be implementable directly from the fan data.","The persistent homology in Example 6.7 may be a combinatorial shadow of derived autoequivalences or of multigraded Castelnuovo–Mumford regularity, and could be studied purely through the alcove subdivision of the section polytope.","One could test whether the same ceiling-truncation construction extends to arbitrary coherent sheaves by applying it to a chosen free resolution; Corollary 6.5 already predicts the resulting homology, so the open question is whether a canonical choice of resolution makes the truncation functor exact in more cases.","The dependence on the incidence function suggests that non-unimodular fans may carry an orientation-type obstruction; comparing two invariant incidence functions on a Picard-rank-two example would isolate whether the main isomorphism is independent of that choice."],"forward_implications":["For each nef divisor $d$, $\\operatorname{trunc}_d(S)$ is resolved in length at most $\\dim X$ by an explicitly described cellular complex, giving a concrete algebraic model for the Fourier–Mukai transform of $\\mathcal O(d)$.","The sheafification of the resolution is exactly $\\mathcal O(d)$, so the complex is a short virtual resolution in the sense that it has no sheaf-level homology beyond the structure sheaf.","The homology of the Fourier–Mukai transform $\\Phi(Q(d))$ for any suitable module $Q$ can be read off from the ceiling-truncated free resolution of $Q$, as stated in Corollary 6.5.","The nontrivial homology exhibited in Example 6.7 shows that exactness can fail at the module level while the sheafified complex is exact, supporting the broader philosophy that allowing irrelevant homology enlarges the class of usable resolutions."],"supporting_citations":[{"why":"Supplies the criterion that a cellular complex attached to a labeled polytopal complex resolves the monomial ideal generated by its vertex labels when the relevant subcomplexes are convex.","marker":"[BS98]"},{"why":"Establishes the correspondence between finitely generated multigraded modules over the total coordinate ring and sheaves on the toric variety, which lets the resolution be interpreted geometrically.","marker":"[Cox95]"},{"why":"Provides the resolution of the diagonal on all smooth projective toric varieties whose Fourier–Mukai transform is reproduced by the paper's cellular complex.","marker":"[HHL24]"},{"why":"Introduces short virtual resolutions for products of projective spaces and the philosophy of allowing irrelevant homology, which the paper extends to all smooth projective toric varieties.","marker":"[BES20]"},{"why":"Gives the unimodular case where lattice points are the only vertices and the construction agrees with the paper's, supplying the polytope subdivision context.","marker":"[BPS01]"},{"why":"Provides the standard vanishing theorem for cohomology of nef line bundles on toric varieties, used in Lemma 3.2 to ensure the spectral sequence degenerates.","marker":"[CLS11]"},{"why":"Supplies the formalism of Fourier–Mukai transforms and the fact that applying a resolution of the diagonal returns a complex quasi-isomorphic to the original sheaf.","marker":"[Huy06]"},{"why":"Provides the lemma on lifting maps between presentations, used to define the ceiling truncation functor on modules.","marker":"[Eis95]"}],"fun_headline_variants":["Explicit short resolutions for all nef toric divisors","Nef truncations on toric varieties get explicit resolutions","Toric truncations resolved: homology can be nontrivial","Short resolutions for nef truncations, via geometry","Ceiling complex yields explicit resolutions for nef truncations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a choice of incidence function can be made globally on the infinite cell complex, invariant under the lattice, so that the boundary maps on the finite cell complex $D$ are exactly the same as the maps in the Fourier–Mukai complex on the quotient torus; the paper checks this locally but not globally, and cells in the quotient can have multiple preimages with identified boundaries.","fun_headline_variants_meta":{"raw":{"variants":["Explicit short resolutions for all nef toric divisors","Nef truncations on toric varieties get explicit resolutions","Toric truncations resolved: homology can be nontrivial","Short resolutions for nef truncations, via geometry","Ceiling complex yields explicit resolutions for nef truncations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1453,"prompt_tokens":840,"completion_tokens":613,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":532}},"tokens_in":456,"tokens_out":613,"duration_ms":5444,"temperature":1.0,"reasoning_tokens":532,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:49:22.592111+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose a smooth toric variety where the quotient cell complex $E$ identifies multiple preimages of a cell (as in the worked example), fix two incidence functions on $C$ that are invariant under translation by $M$, compute the boundary maps of $F(d)_\\bullet$ for both, and compare their homologies; if the homologies differ, or if $\\partial^2\\neq 0$ for either one, the central resolution and isomorphism claims fail.","supporting_citations":[],"review_version":1}