arxiv: 2605.01504 · v1 · submitted 2026-05-02 · 🧮 math.AG

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Equivariant sheaves on toric prevarieties

Jyoti Dasgupta, Kartik Roy

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Pith reviewed 2026-05-09 18:04 UTC · model grok-4.3

classification 🧮 math.AG

keywords toric prevarietiesequivariant sheavesquasicoherent sheavescombinatorial descriptionfan datatorus actionsalgebraic geometry

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The pith

Equivariant quasicoherent sheaves on toric prevarieties admit a combinatorial description via assignments on cones.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that equivariant quasicoherent sheaves on toric prevarieties correspond to collections of vector spaces on the cones of the fan together with linear maps for inclusions of faces. Toric prevarieties are obtained by gluing affine toric varieties along opens that need not satisfy the separation axiom. This lets one encode the sheaves using only local linear-algebra data without separate global gluing conditions. A reader cares because the same data suffices for both separated and non-separated cases, simplifying calculations of global sections and morphisms.

Core claim

Equivariant quasicoherent sheaves on toric prevarieties are described by assigning a vector space to each cone together with linear maps for every face inclusion such that the assignments satisfy the natural compatibility relations on the overlaps, and this correspondence remains bijective even when the underlying prevariety is non-separated.

What carries the argument

Cone-by-cone linear-algebra data: a vector space on each cone of the fan with linear maps along face relations.

If this is right

Morphisms between such sheaves correspond to collections of linear maps between the assigned vector spaces that commute with the face maps.
The category of equivariant quasicoherent sheaves is equivalent to a category of combinatorial objects built directly from the fan.
Global sections of an equivariant sheaf can be read off as the subspace of the vector space on the maximal cone that is fixed by the face maps.
The description applies uniformly whether or not the prevariety satisfies the Hausdorff axiom.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same combinatorial objects may classify equivariant sheaves on other non-separated toric spaces obtained by more general gluings.
Cohomology groups of these sheaves become computable by linear algebra on the vector spaces attached to the cones.
Examples of indecomposable sheaves can be constructed by choosing vector spaces of varying dimensions on the cones and solving for compatible maps.

Load-bearing premise

The non-separated gluings of affine pieces in a toric prevariety do not prevent the cone-by-cone vector-space assignments from determining unique global sheaves.

What would settle it

An explicit toric prevariety together with an equivariant quasicoherent sheaf whose local data on cones cannot be assembled into any single collection of vector spaces and face maps satisfying the compatibility rules.

Figures

Figures reproduced from arXiv: 2605.01504 by Jyoti Dasgupta, Kartik Roy.

**Figure 1.** Figure 1: Maximal invariant affine open chart of the toric prevariety The tangent bundle TX is given by the following data: T [ρ1,1](s) =    0 s ≤ −2 Span(0, 1) s = −1 k 2 s ≥ 0, , T [ρ2,1](s) =    0 s ≤ −2 Span(1, −1) s = −1 k 2 s ≥ 0, , T [ρ3,2](s) =    0 s ≤ −2 Span(1, 0) s = −1 k 2 s ≥ 0, and T [ρ4,2](s) =    0 s ≤ −2 Span(−1, −1) s = −1 k 2 s ≥ 0. By Remark 4.2, the tangent bundle does not split. Ex… view at source ↗

**Figure 2.** Figure 2: Affine system of fans describing the toric prevariety in Example 5.3 The indexing set is given by Λ = {[ρ1, 1], [ρ2, 1], [ρ3, 2], [ρ4, 4]}, where ρ1 = R≥0(1, 0), ρ2 = R≥0(1, 1), ρ3 = R≥0(0, 1) and ρ4 = R≥0(−1, −1) (see view at source ↗

read the original abstract

Toric prevarieties are non-separated analogues of toric varieties. Perling \cite{Perling_equivariant_sheaves_tor_var} provided a combinatorial description of equivariant quasicoherent sheaves on toric varieties, extending earlier ideas of Klyachko, who outlined a general framework for equivariant torsion free sheaves in an unpublished work \cite{kly_sheaf}. In this article, we present a combinatorial description of equivariant quasicoherent sheaves on toric prevarieties.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Perling's description to prevarieties but non-separated gluings may need extra compatibility checks.

read the letter

The main thing to know is that the paper supplies a combinatorial description of equivariant quasicoherent sheaves on toric prevarieties, adapting the cone-by-cone linear-algebra data that Perling used for the separated case of toric varieties. It positions this as the first such description for the non-separated setting, building on Klyachko's earlier outline as well. The claim is that the same filtrations or graded modules with face compatibilities classify the sheaves even when the gluings are not separated. This is a direct extension rather than a major reorganization of the field. The authors do a clean job of stating the result up front and citing the prior work without unnecessary machinery. The combinatorial style stays consistent with what has worked in toric geometry before. The potential soft spot is exactly the gluing issue raised in the stress test. Non-separated identifications between cones can impose extra conditions on restriction maps or stabilizer actions that the standard face-relation data may not automatically satisfy. The abstract does not spell out whether an additional cocycle-type condition is needed or whether the existing data suffices without change. If the full text does not isolate and verify this point with examples or a proof step, that would be a gap worth fixing. The paper is aimed at specialists already working in toric algebraic geometry and equivariant sheaves. A reader familiar with Perling's paper will see the value immediately and can check the details themselves. It deserves a serious referee because the claim is precise and the extension is natural. I would send it out for review and ask the referees to focus on how the non-separated loci are handled in the combinatorial data.

Referee Report

1 major / 1 minor

Summary. The manuscript claims to extend Perling's combinatorial classification of equivariant quasicoherent sheaves on toric varieties (via cone-by-cone filtrations or graded modules with face compatibilities) to the non-separated setting of toric prevarieties, asserting that the same linear-algebra data suffices without additional conditions.

Significance. A correct extension would be a modest but useful advance in toric geometry, allowing explicit descriptions of sheaves on spaces that arise naturally in quotients and gluings; it builds directly on cited prior work by Perling and Klyachko and could facilitate computations in equivariant settings.

major comments (1)

[Main result (likely §3 or the statement following the abstract)] The central claim that Perling's cone-by-cone data extends directly is load-bearing, yet the manuscript does not isolate or verify the compatibility of restriction maps and stabilizer actions on non-separated gluing loci. If two cones are identified along a non-separated subvariety, the induced maps on the associated modules may fail to satisfy the required cocycle condition unless extra data is imposed; no explicit check or counter-example ruling out this obstruction appears in the main theorem or its proof.

minor comments (1)

[Introduction] The abstract and introduction cite Perling and Klyachko but do not clarify which parts of the separated-case construction are copied verbatim versus modified; a short comparison table or paragraph would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a potential subtlety in extending the combinatorial description to the non-separated setting. We address the major comment below and will incorporate clarifications in the revised manuscript.

read point-by-point responses

Referee: The central claim that Perling's cone-by-cone data extends directly is load-bearing, yet the manuscript does not isolate or verify the compatibility of restriction maps and stabilizer actions on non-separated gluing loci. If two cones are identified along a non-separated subvariety, the induced maps on the associated modules may fail to satisfy the required cocycle condition unless extra data is imposed; no explicit check or counter-example ruling out this obstruction appears in the main theorem or its proof.

Authors: We appreciate the referee drawing attention to the gluing loci. In the manuscript the combinatorial data (modules on cones with face-restriction maps satisfying the same compatibilities as in Perling) is defined directly on the fan of the prevariety; the non-separated gluings are already encoded in the fan structure itself. The equivalence proof constructs the sheaf by gluing the affine pieces and verifies that the resulting object is quasicoherent and equivariant precisely because the restriction maps on overlaps (including non-separated ones) are required to be torus-equivariant and to satisfy the face relations by definition of the data. Consequently the cocycle condition on triple overlaps and on identified loci holds automatically from the consistency of the linear-algebra data; no supplementary cocycle data is needed. To make this verification explicit we will insert a short paragraph immediately after the statement of the main theorem that isolates the restriction maps on non-separated loci and confirms the cocycle identity using only the given face compatibilities. We therefore maintain that the same data suffices, but agree that an isolated check improves readability. revision: partial

Circularity Check

0 steps flagged

No circularity; direct extension of independent prior combinatorial classification

full rationale

The paper cites Perling's combinatorial description for the separated toric variety case and Klyachko's framework as external inputs, then claims to extend the same cone-by-cone linear-algebra data (filtrations with face compatibilities) to the non-separated prevariety setting. No equation or definition in the provided abstract reduces the new claim to a fitted parameter, self-referential renaming, or self-citation chain. The central result is presented as a verification that the existing data suffices under the weaker gluing conditions of prevarieties, which is an independent mathematical step rather than a tautology. This is the expected non-circular outcome for an extension paper whose load-bearing citations are to non-overlapping prior authors.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definitions of toric prevarieties and equivariant sheaves together with the combinatorial framework already developed by Perling and Klyachko; no new free parameters or invented entities are indicated in the abstract.

axioms (2)

domain assumption Toric prevarieties admit a fan-like combinatorial description analogous to that of toric varieties.
Invoked when extending Perling's classification to the non-separated setting.
domain assumption Equivariant quasicoherent sheaves on these spaces are determined by linear-algebra data on the cones of the fan.
Core assumption inherited from the cited works of Perling and Klyachko.

pith-pipeline@v0.9.0 · 5374 in / 1450 out tokens · 50148 ms · 2026-05-09T18:04:56.452332+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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