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arxiv: 2604.23591 · v1 · submitted 2026-04-26 · 🧮 math.AG

Exceptional loci of F-blowups and G-Hilbert schemes

Enrique Ch\'avez-Mart\'inez , Yutaro Kaijima , Takehiko Yasuda This is my paper

Pith reviewed 2026-05-08 05:25 UTC · model grok-4.3

classification 🧮 math.AG MSC 14E1514M25
keywords F-blowupsG-Hilbert schemestoric varietiesexceptional lociessential divisorsprime divisorsQ-factorial
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The pith

A combinatorial formula gives the dimension of the center of any prime divisor on the F-blowup of a normal toric variety.

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

The paper works out how to read the dimension of the center of a prime divisor on the F-blowup directly from combinatorial data attached to a normal toric variety. This matters because F-blowups serve as a toric model for the G-Hilbert scheme when the variety is Q-factorial, and the same data also controls the exceptional loci that appear in resolutions of singularities. The authors supply both the explicit formula and a working algorithm that computes the dimension, then examine how these centers relate to essential divisors over three-dimensional terminal and canonical singularities. They close with a simple numerical condition on the data that forces the center to have positive dimension.

Core claim

For a normal toric variety, the dimension of the center of a prime divisor on its F-blowup is determined by combinatorial data coming from the fan or the lattice points; an algorithm extracts this number from the data. When the variety is Q-factorial the same description applies to the exceptional loci of the corresponding G-Hilbert scheme. The construction is compared with essential divisors in the three-dimensional terminal and canonical cases, and a simple inequality on the combinatorial invariants guarantees that the center has positive dimension.

What carries the argument

The combinatorial formula that expresses the dimension of the center of a prime divisor on the F-blowup in terms of lattice or fan data of the toric variety, together with the algorithm that evaluates it.

If this is right

  • In the Q-factorial case the same formula describes the exceptional loci of the G-Hilbert scheme.
  • For three-dimensional terminal and canonical singularities the centers on the F-blowup are compared directly with essential divisors.
  • A simple numerical test on the combinatorial data detects which prime divisors acquire positive-dimensional centers on the F-blowup.

Where Pith is reading between the lines

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

  • The formula may let one decide positivity of centers without constructing the full F-blowup geometry.
  • Similar lattice counting could be tested on non-toric varieties that admit a toric degeneration.
  • The algorithm supplies a concrete way to list all positive-dimensional centers once the fan is given.

Load-bearing premise

The input varieties are normal toric varieties, possibly Q-factorial.

What would settle it

An explicit prime divisor over a normal toric variety whose center on the F-blowup has a dimension that differs from the value predicted by the combinatorial formula.

Figures

Figures reproduced from arXiv: 2604.23591 by Enrique Ch\'avez-Mart\'inez, Takehiko Yasuda, Yutaro Kaijima.

Figure 2
Figure 2. Figure 2: (δ w 1 ) ∨ Definition 4.6. Let ci,w = c := min{(u, −vi) | u ∈ (δ w i ) ∨ ∩ M \ w ⊥} ∈ R>0. Then, we define sets Critw i ⊂ M as Critw i := {u ∈ (δ w i ) ∨ ∩ M \ {0} | (u, −vi) < ci,w} ⊂ w ⊥. We also define Critw := [ 1≤i≤d vi∈µ Critw i . Proposition 4.7. Every element of Critw i is the vector associated to some w￾critical arrow in AR. Conversely, every w-critical arrow with tail on the ray R≥0ui has the ass… view at source ↗
read the original abstract

We study exceptional loci of F-blowups of normal toric varieties. In the $\Q$-factorial case, this study amounts to studying the exceptional loci of $G$-Hilbert schemes. We give a formula for the dimension of the center of a prime divisor on the F-blowup in terms of combinatorial data, together with an algorithm for computing it. Moreover, we study the relation between F-blowups and essential divisors for three-dimensional terminal singularities and canonical singularities. Finally, we give a simple condition ensuring that a prime divisor over the given toric variety has a positive-dimensional center on the F-blowup.

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.

Referee Report

0 major / 2 minor

Summary. The paper studies exceptional loci of F-blowups of normal toric varieties. In the Q-factorial case this reduces to the exceptional loci of G-Hilbert schemes. It supplies a formula, expressed in combinatorial data attached to the fan, for the dimension of the center of any prime divisor on the F-blowup, together with an explicit algorithm to compute that dimension. The authors also examine the relation between these loci and essential divisors in the setting of three-dimensional terminal and canonical singularities, and they give a simple positivity criterion guaranteeing that a prime divisor has positive-dimensional center on the F-blowup.

Significance. If the stated combinatorial formula and algorithm are correct, the work supplies a practical, explicitly computable bridge between the geometry of F-blowups (and G-Hilbert schemes) and the combinatorial data of toric fans. This is a genuine advance for explicit calculations in birational geometry of toric varieties. The additional results on essential divisors in dimension three and the positivity criterion are useful specializations that connect the general theory to well-studied classes of singularities.

minor comments (2)
  1. The algorithm is described in general terms; inserting a fully worked numerical example (for instance on a simple weighted projective 3-space or a toric surface) would make the steps immediately verifiable and would strengthen the exposition.
  2. Notation for the combinatorial invariants (rays, cones, multiplicities) used in the dimension formula is introduced piecemeal; a short table or diagram summarizing the input data and the output dimension would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation and recommendation of minor revision. The summary provided accurately describes our results on the exceptional loci of F-blowups of normal toric varieties, the reduction to G-Hilbert schemes in the Q-factorial case, the combinatorial formula and algorithm for the dimension of centers of prime divisors, as well as the connections to essential divisors in dimension three and the positivity criterion. As no specific major comments were raised, we have no revisions to propose at this stage but remain available for any clarifications or minor adjustments.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained combinatorial geometry

full rationale

The paper derives a dimension formula and algorithm for centers of prime divisors on F-blowups of normal toric varieties (reducing to G-Hilbert exceptional loci when Q-factorial) directly from the fan/ray description of divisors and centers supplied by toric geometry. No step equates a claimed prediction to a fitted input by construction, invokes a self-citation as the sole justification for a uniqueness or ansatz claim, or renames a known pattern as new unification. The positivity criterion and 3-fold terminal/canonical comparisons are likewise explicit combinatorial checks resting on standard birational geometry, with all inputs externally verifiable outside the paper's own fitted values or prior self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard domain assumptions from toric geometry and birational geometry; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Normal toric varieties admit combinatorial descriptions via fans and lattice data.
    Invoked as the basis for expressing the dimension formula combinatorially.
  • domain assumption F-blowups and G-Hilbert schemes are well-defined in the context of Q-factorial toric varieties.
    Used to equate the study of exceptional loci in the Q-factorial case.

pith-pipeline@v0.9.0 · 5410 in / 1205 out tokens · 17490 ms · 2026-05-08T05:25:33.820746+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

2 extracted references

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    [Sat25] Yusuke Sato

    [Sag26] The Sage Developers.SageMath, the Sage Mathematics Software System (Version 4.4.2), 2026.https://www.sagemath.org. [Sat25] Yusuke Sato. Complete coprime cyclic quotient singularities.Kyoto J. Math., 65(3):487–502,

    2026

  2. [2]

    [TY09] Yukinobu Toda and Takehiko Yasuda

    ©2023. [TY09] Yukinobu Toda and Takehiko Yasuda. Noncommutative resolution,F-blowups and D-modules.Adv. Math., 222(1):318–330,

    2023