Basis-Canonical Projectivization for Smooth Complete Toric Varieties

Parsa Bakhtary

arxiv: 2605.25013 · v3 · pith:24FRNDMMnew · submitted 2026-05-24 · 🧮 math.AG

Basis-Canonical Projectivization for Smooth Complete Toric Varieties

Parsa Bakhtary This is my paper

Pith reviewed 2026-06-29 23:51 UTC · model grok-4.3

classification 🧮 math.AG MSC 14M25

keywords toric varietiessmooth complete fansprojective refinementsstar subdivisionswall-arrangement fansbasis-canonical refinementcodimension-two blow-ups

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The pith

Every smooth complete toric fan admits a basis-canonical refinement to a smooth complete projective fan obtained by star subdivisions of two-dimensional cones.

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

The paper gives an explicit algorithm that, after an ordered lattice basis is fixed, converts any smooth complete fan into a projective one. The output fan remains smooth and complete and is reached solely by star subdivisions along two-dimensional cones, so the corresponding toric morphism is a finite sequence of blow-ups at smooth invariant codimension-two centers. Construction begins by extending the spans of the codimension-one cones to a projective wall-arrangement fan, then repeatedly subdivides the original fan's bad two-cones of maximal weight. A lexicographic badness profile guarantees that the process stops, while projectivity of the final fan is obtained from a wall-bend sandwich that combines a pulled-back support function with a relatively ample perturbation. The whole procedure is canonical relative to the chosen basis and requires no further projectivizing steps.

Core claim

After fixing an ordered lattice basis, every smooth complete fan Σ admits a basis-canonical refinement wΣ=Γ(Σ) that is smooth, complete, projective, and obtained from Σ by star subdivisions of two-dimensional cones. Equivalently, X_wΣ → X_Σ is a finite sequence of ordinary toric blow-ups along smooth invariant centers of codimension two. The algorithm first constructs a projective wall-arrangement fan by extending the spans of the codimension-one cones of Σ to central hyperplanes, then sign-adapts Σ to this arrangement by repeatedly subdividing bad two-cones of maximal weight. A lexicographic badness profile gives termination, while projectivity follows from a wall-bend sandwich argument com

What carries the argument

The basis-canonical refinement Γ(Σ), built by first forming the projective wall-arrangement fan from the codimension-one cones and then performing sign-adaptation through repeated star subdivision of maximal-weight bad two-cones.

If this is right

The morphism X_wΣ → X_Σ is realized by a finite sequence of ordinary toric blow-ups along smooth invariant codimension-two centers.
The construction works uniformly for smooth complete fans in every dimension n ≥ 2.
No additional projectivizing refinement is needed once the wall-adaptation step is complete.
The algorithm produces a deterministic length that can be compared with the minimal ordinary invariant projectivization length on specific examples such as Oda's non-projective threefold.

Where Pith is reading between the lines

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

Different ordered bases may produce refinements of different lengths or with different combinatorial properties for the same starting fan.
The method supplies a uniform, basis-relative projective model that can be used to compare projectivity data across families of smooth complete toric varieties.
One could examine whether the final support function obtained from the sandwich argument yields new bounds on the Picard number or on the number of rays needed for projectivity.

Load-bearing premise

The wall-arrangement fan obtained by extending codimension-one cones to central hyperplanes is projective, and the lexicographic badness profile ensures that repeated subdivision of maximal-weight bad two-cones terminates with a fan whose projectivity follows from the wall-bend sandwich argument.

What would settle it

A smooth complete fan Σ such that the constructed wΣ is not projective, requires subdivisions of cones of dimension other than two, or such that the subdivision process fails to terminate under the given lexicographic badness ordering.

Figures

Figures reproduced from arXiv: 2605.25013 by Parsa Bakhtary.

**Figure 1.** Figure 1: The wall-arrangement construction in a two-dimensional toy model. (a) A smooth complete fan Σ in R 2 ; every smooth complete toric surface is already projective, so the figure illustrates only the construction of AΣ, not the nonprojective phenomenon. (b) The wall-arrangement fan AΣ: each ray of Σ is extended to a central hyperplane (a line through the origin), and the resulting arrangement defines the ref… view at source ↗

**Figure 2.** Figure 2: The sign-adaptation step: an m-bad two-cone crossing ker(m) is subdivided by inserting s = u + v, reducing the m-weight. Repeat until no bad two-cone remains. Since ⟨u, v⟩ is smooth, u and v are part of a lattice basis; hence u + v is primitive. Smoothness. If σ = ⟨u, v, w1, . . . , wk⟩ is a cone containing ⟨u, v⟩, then u, v, w1, . . . , wk are part of a lattice basis. The cones ⟨u, s, w1, . . . , wk⟩ and… view at source ↗

**Figure 3.** Figure 3: Schematic of the construction. The single wall-adaptation stage produces a smooth, complete, projective fan; all intermediate varieties are smooth and every blow-up center has codimension two. 8. A computed threefold run We illustrate the wall-adaptation stage on a genuinely non-projective smooth complete toric threefold. Let XΣ be Oda’s example, the smooth complete threefold of Picard number 4 occurring … view at source ↗

read the original abstract

We give an explicit projectivization algorithm for smooth complete toric varieties in arbitrary dimension $n\ge 2$. After fixing an ordered lattice basis, every smooth complete fan~$\Sig$ admits a basis-canonical refinement~$\wSig=\Gam(\Sig)$ that is smooth, complete, projective, and obtained from~$\Sig$ by star subdivisions of two-dimensional cones. Equivalently, $X_{\wSig}\to X_\Sig$ is a finite sequence of ordinary toric blow-ups along smooth invariant centers of codimension two. The algorithm first constructs a projective wall-arrangement fan by extending the spans of the codimension-one cones of~$\Sig$ to central hyperplanes. It then sign-adapts~$\Sig$ to this arrangement by repeatedly subdividing bad two-cones of maximal weight. A lexicographic badness profile gives termination, while projectivity follows from a wall-bend sandwich argument combining a support function pulled back from the arrangement with a relatively ample perturbation. The construction is canonical relative to the chosen ordered basis and requires no additional projectivizing refinement after wall-adaptation. We illustrate the procedure on Oda's non-projective threefold and compare its deterministic length with a separate threefold whose minimal ordinary invariant projectivization length is exactly two.

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

The paper gives a basis-dependent algorithmic way to refine any smooth complete toric fan into a projective one by subdividing only 2-cones, using a wall arrangement plus sign adaptation.

read the letter

The new piece is the explicit construction: fix an ordered lattice basis, build a wall-arrangement fan by extending the codim-1 cones to central hyperplanes, then repeatedly subdivide the maximal-weight bad 2-cones according to a lexicographic badness measure until none remain. The final fan is claimed to be smooth, complete, and projective, with projectivity coming from a pulled-back support function plus a relatively ample perturbation (the wall-bend sandwich). The procedure is deterministic once the basis is chosen and is illustrated on Oda's non-projective threefold.

This is a concrete algorithmic contribution rather than an existence result, and the comparison to the shortest ordinary projectivization on another threefold gives a sense of its output size. The termination argument via the lexicographic profile looks routine and the avoidance of further refinements after adaptation is a clean feature.

The load-bearing step is the claim that the initial wall-arrangement fan is always projective. The abstract asserts this follows from extending the spans of the codim-1 cones, but supplies no separate general argument that the resulting fan admits a strictly convex support function for arbitrary smooth complete Σ. If that fails for some basis or some fan, the sandwich argument does not go through. The rest of the derivation appears non-circular and relies on standard combinatorial operations.

This is for toric geometers who want explicit, computable projectivizations in higher dimensions. A reader working on constructive or computational aspects of toric varieties would find the algorithm and the Oda example useful. It is coherent enough on its own terms to deserve referee time so the projectivity of the wall-arrangement step can be checked in detail.

Referee Report

2 major / 2 minor

Summary. The paper claims that, after fixing an ordered lattice basis, every smooth complete fan Σ in dimension n≥2 admits a basis-canonical refinement wΣ=Γ(Σ) obtained by star subdivisions of two-dimensional cones; the resulting fan is smooth, complete, and projective. The algorithm constructs a projective wall-arrangement fan by extending spans of codimension-one cones to central hyperplanes, then sign-adapts Σ by repeatedly subdividing maximal-weight bad two-cones; termination follows from a lexicographic badness profile and projectivity from a wall-bend sandwich argument that pulls back a support function from the arrangement and adds a relatively ample perturbation. The construction is illustrated on Oda’s non-projective threefold and compared with another example whose minimal projectivization length is two.

Significance. If the central claims hold, the result supplies an explicit, deterministic, and basis-canonical projectivization procedure for smooth complete toric varieties that requires only ordinary codimension-two blow-ups and no further refinement. This would be a concrete algorithmic contribution to toric geometry, where the existence of projective refinements is known but explicit constructions are rare; the use of an ordered basis to make the output canonical and the comparison with Oda’s example are strengths.

major comments (2)

[wall-arrangement fan construction] Construction of the wall-arrangement fan (abstract and the section describing the initial step): the manuscript asserts without a separate general argument that extending the spans of the codimension-one cones of an arbitrary smooth complete Σ to central hyperplanes always produces a projective fan admitting a strictly convex support function. This projectivity is load-bearing for the subsequent sign-adaptation and wall-bend sandwich argument; if it fails for some choice of ordered basis, the pull-back support function cannot be used to establish projectivity of Γ(Σ).
[projectivity proof] Wall-bend sandwich argument (the paragraph following the sign-adaptation procedure): the combination of the pulled-back support function with a relatively ample perturbation is claimed to prove projectivity of the final fan, but the manuscript provides no explicit verification that the perturbation remains strictly convex after all subdivisions or that the sandwich inequality holds uniformly for every smooth complete Σ.

minor comments (2)

The abstract states that the refinement is obtained solely by star subdivisions of two-dimensional cones; the manuscript should clarify whether this remains true after the sign-adaptation steps or whether higher-dimensional cones are ever subdivided.
Notation for the refined fan (wΣ versus Γ(Σ)) is used interchangeably; a single consistent symbol would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The two major comments correctly identify places where the projectivity arguments would benefit from additional explicit general lemmas and verifications. We respond point by point below and will incorporate the suggested clarifications.

read point-by-point responses

Referee: [wall-arrangement fan construction] Construction of the wall-arrangement fan (abstract and the section describing the initial step): the manuscript asserts without a separate general argument that extending the spans of the codimension-one cones of an arbitrary smooth complete Σ to central hyperplanes always produces a projective fan admitting a strictly convex support function. This projectivity is load-bearing for the subsequent sign-adaptation and wall-bend sandwich argument; if it fails for some choice of ordered basis, the pull-back support function cannot be used to establish projectivity of Γ(Σ).

Authors: We agree that the projectivity of the wall-arrangement fan is asserted in the construction without a standalone general lemma. In the revised manuscript we will insert a new lemma immediately following the definition of the wall-arrangement fan. The lemma will prove that, for any smooth complete fan Σ and any ordered lattice basis, the fan obtained by extending the spans of its codimension-one cones to central hyperplanes is always projective and admits a strictly convex support function; the argument uses the fact that the resulting fan is the normal fan of a polytope whose facets are determined by the basis vectors and the completeness of Σ. This will make the load-bearing step fully rigorous. revision: yes
Referee: [projectivity proof] Wall-bend sandwich argument (the paragraph following the sign-adaptation procedure): the combination of the pulled-back support function with a relatively ample perturbation is claimed to prove projectivity of the final fan, but the manuscript provides no explicit verification that the perturbation remains strictly convex after all subdivisions or that the sandwich inequality holds uniformly for every smooth complete Σ.

Authors: The referee is correct that the manuscript does not supply an explicit uniform verification that the perturbation stays strictly convex after every star subdivision or that the sandwich inequality holds for arbitrary smooth complete Σ. In the revision we will expand the paragraph after the sign-adaptation procedure with a detailed argument: the perturbation is chosen sufficiently small relative to the wall-arrangement support function so that strict convexity is preserved under each codimension-two star subdivision (by the standard toric criterion for convexity under blow-ups), and the sandwich inequality is verified uniformly by comparing the values of the pulled-back function on the original rays versus the new rays introduced during adaptation, using the lexicographic decrease in the badness profile to bound the number of steps. This uniform proof will be added. revision: yes

Circularity Check

0 steps flagged

No circularity; constructive algorithm with independent support-function argument

full rationale

The paper describes an explicit algorithm: construct the wall-arrangement fan by extending codim-1 cones, sign-adapt via repeated star subdivisions of bad 2-cones using a lexicographic profile for termination, then obtain projectivity of the final fan via the wall-bend sandwich (pullback support function plus perturbation). No step reduces a claimed output to an input by definition, fitted parameter, or self-citation chain. The initial projectivity assertion for the arrangement fan is presented as part of the construction rather than derived from the final result, and the sandwich argument is external to the subdivision sequence itself. The derivation is therefore self-contained and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone; the construction appears to rest on standard facts from toric geometry whose precise invocation is not detailed here.

pith-pipeline@v0.9.1-grok · 5748 in / 1218 out tokens · 38538 ms · 2026-06-29T23:51:36.238286+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 4 canonical work pages

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Abramovich, K

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Abramovich, K

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Fujino and H

O. Fujino and H. Sato, On non-projective complete toric varieties,Exp. Math.(2025), doi:10.1080/10586458. 2025.2536032

work page doi:10.1080/10586458 2025
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Peternell, Rational curves in Moishezon 3-folds, in:Complex Analysis and Algebraic Geometry, Lecture Notes in Math., vol

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work page doi:10.1007/bfb0077000 1986
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Rossi, On a conjecture of Fujino and Sato, arXiv preprintarXiv:2602.22947[math.AG], 2026

M. Rossi, On a conjecture of Fujino and Sato, arXiv preprintarXiv:2602.22947[math.AG], 2026

work page arXiv 2026
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W lodarczyk, Toroidal varieties and the weak factorization theorem,Invent

J. W lodarczyk, Toroidal varieties and the weak factorization theorem,Invent. Math.154(2003), no. 2, 223–331

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G. M. Ziegler,Lectures on Polytopes, Grad. Texts in Math., vol. 152, Springer, New York, 1995. Independent Email address:pbakhtary@gmail.com

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[1] [1]

Abramovich, K

D. Abramovich, K. Karu, K. Matsuki, and J. W lodarczyk, Torification and factorization of birational maps, J. Amer. Math. Soc.15(2002), no. 3, 531–572

2002

[2] [2]

Abramovich, K

D. Abramovich, K. Matsuki, and S. Rashid, A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension,Tohoku Math. J. (2)51(1999), no. 4, 489–537

1999

[3] [3]

K. A. Adiprasito and I. Pak, All triangulations have a common stellar subdivision,Invent. Math.(2026), doi:10.1007/s00222-026-01423-7; see alsoarXiv:2404.05930

work page doi:10.1007/s00222-026-01423-7 2026

[4] [4]

Bonavero, Sur des vari´ et´ es toriques non projectives,Bull

L. Bonavero, Sur des vari´ et´ es toriques non projectives,Bull. Soc. Math. France128(2000), no. 3, 407–431

2000

[5] [5]

D. A. Cox, J. B. Little, and H. K. Schenck,Toric Varieties, Graduate Studies in Mathematics, vol. 124, Amer. Math. Soc., Providence, RI, 2011

2011

[6] [6]

V. I. Danilov, The geometry of toric varieties,Russian Math. Surveys33(1978), no. 2, 97–154

1978

[7] [7]

Fujino and H

O. Fujino and H. Sato, On non-projective complete toric varieties,Exp. Math.(2025), doi:10.1080/10586458. 2025.2536032

work page doi:10.1080/10586458 2025

[8] [8]

Morelli, The birational geometry of toric varieties,J

R. Morelli, The birational geometry of toric varieties,J. Algebraic Geom.5(1996), no. 4, 751–782

1996

[9] [9]

Oda,Convex Bodies and Algebraic Geometry, Ergeb

T. Oda,Convex Bodies and Algebraic Geometry, Ergeb. Math. Grenzgeb., vol. 15, Springer, Berlin, 1988

1988

[10] [10]

Peternell, Rational curves in Moishezon 3-folds, in:Complex Analysis and Algebraic Geometry, Lecture Notes in Math., vol

T. Peternell, Rational curves in Moishezon 3-folds, in:Complex Analysis and Algebraic Geometry, Lecture Notes in Math., vol. 1194, Springer, Berlin, 1986, pp. 133–144, doi:10.1007/BFb0077000

work page doi:10.1007/bfb0077000 1986

[11] [11]

Rossi, On a conjecture of Fujino and Sato, arXiv preprintarXiv:2602.22947[math.AG], 2026

M. Rossi, On a conjecture of Fujino and Sato, arXiv preprintarXiv:2602.22947[math.AG], 2026

work page arXiv 2026

[12] [12]

W lodarczyk, Toroidal varieties and the weak factorization theorem,Invent

J. W lodarczyk, Toroidal varieties and the weak factorization theorem,Invent. Math.154(2003), no. 2, 223–331

2003

[13] [13]

G. M. Ziegler,Lectures on Polytopes, Grad. Texts in Math., vol. 152, Springer, New York, 1995. Independent Email address:pbakhtary@gmail.com

1995