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arxiv: 2606.17884 · v2 · pith:GGPNSJLVnew · submitted 2026-06-16 · 🧮 math.AG

Naive atoms of blowups: examples

Christian B\"ohning , Hans-Christian Graf von Bothmer , Zac Su'a This is my paper

Pith reviewed 2026-06-26 22:48 UTC · model grok-4.3

classification 🧮 math.AG
keywords naive atomic decompositionsIritani's blowup formulasmooth projective varietiesblowupsalgebraic geometryexamplesdecompositions
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The pith

Naive atomic decompositions of smooth projective varieties satisfy a naive version of Iritani's blowup formula in several examples.

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

The paper defines naive atomic decompositions of smooth projective varieties. It shows that these decompositions satisfy a naive version of Iritani's blowup formula in several examples. The examples are complicated enough to show most interesting features of the general theory but simple enough to be computable by elementary methods. This matters because it provides concrete, verifiable instances where the decompositions behave predictably under blowups, allowing readers to see the mechanism at work without advanced machinery.

Core claim

We define naive atomic decompositions of smooth projective varieties. We show that they satisfy a naive version of Iritani's blowup formula in several examples that are complicated enough to show most interesting features of the general theory while being simple enough to be computable by elementary methods.

What carries the argument

Naive atomic decompositions, a decomposition of the variety into atomic parts that permits application of a simplified blowup formula.

If this is right

  • The formula holds in examples involving blowups of points and other subvarieties.
  • Elementary methods suffice to verify the property in non-trivial cases.
  • The decompositions capture the main features of the blowup behavior.
  • Such examples serve as a testing ground for the general theory.

Where Pith is reading between the lines

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

  • Generalizing to all smooth projective varieties might be possible if the examples are typical.
  • This could simplify computations of invariants in blown-up varieties.
  • Similar naive approaches might apply to other formulas in algebraic geometry.
  • Testing more examples could reveal if the naive version is always sufficient.

Load-bearing premise

The selected examples are representative of the general theory and capture its essential features without hidden dependencies on advanced tools.

What would settle it

Computing the naive atomic decomposition and checking the blowup formula for one of the paper's examples and finding a mismatch would falsify the claim.

read the original abstract

We define naive atomic decompositions of smooth projective varieties. We show that they satisfy a naive version of Iritani's blowup formula in several examples that are complicated enough to show most interesting features of the general theory while being simple enough to be computable by elementary methods.

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 / 3 minor

Summary. The manuscript defines naive atomic decompositions of smooth projective varieties and verifies, via explicit elementary computations, that these decompositions satisfy a naive version of Iritani's blowup formula in several examples chosen to illustrate most interesting features while remaining computable.

Significance. If the example verifications hold, the work supplies concrete, machine-checkable data points on the behavior of atomic decompositions under blowups. The elementary character of the computations is a positive feature, as it allows direct inspection without advanced tools and may serve as a test bed for any future general statement of the naive formula.

minor comments (3)
  1. The abstract refers to 'several examples' without naming the varieties or the blowup centers; adding this information would immediately clarify the scope for readers.
  2. The definition of naive atomic decompositions appears in the body without a highlighted statement or numbered definition; extracting it into a standalone definition would improve readability.
  3. It is unclear whether the paper includes a table or summary comparing the naive decompositions before and after each blowup; such a table would make the verification easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition that the elementary computations provide machine-checkable data points, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit example computations

full rationale

The paper defines naive atomic decompositions of smooth projective varieties and verifies that they satisfy a naive version of Iritani's blowup formula via direct, elementary computations in a small number of explicitly chosen examples. No general theorem is claimed, no parameters are fitted to data and then repurposed as predictions, and no load-bearing steps reduce to self-citations, self-definitions, or imported ansatzes. The derivation chain consists of concrete calculations whose outcomes are stated as observed facts rather than forced by construction from the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities beyond the core definition itself. The 'naive atomic decomposition' is introduced as a new concept whose properties are then checked in examples.

invented entities (1)
  • naive atomic decompositions no independent evidence
    purpose: To provide a simplified decomposition of smooth projective varieties that obeys a naive blowup formula
    Introduced in the paper as the central new object; no independent evidence outside the examples is mentioned in the abstract.

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discussion (0)

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Reference graph

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33 extracted references · 1 canonical work pages

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