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arxiv: 2606.30160 · v1 · pith:6HMELZZAnew · submitted 2026-06-29 · 🧮 math.AG · math.AC· math.CO

On Property N_p of line bundles on smooth projective toric varieties

Lei Song , Huanqi Wen This is my paper

Pith reviewed 2026-06-30 03:49 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CO
keywords Property N_pline bundlestoric varietiesuniform unimodularityThomsen stratificationintersection numbersprojective varietiessyzygies
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The pith

If a smooth projective toric variety meets uniform unimodularity and Thomsen stratification conditions, line bundles with L·C ≥ n-1+p on invariant curves satisfy Property N_p.

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

The paper proves a criterion for when line bundles on smooth projective toric varieties have Property N_p. It requires the variety to satisfy uniform unimodularity and a Thomsen stratification intersection-number condition. Under these assumptions, a lower bound on the intersection number of the line bundle with every torus-invariant curve guarantees the property. The conditions are shown to hold for several families of toric varieties and to be preserved when taking finite products of such varieties.

Core claim

If a smooth projective toric variety X of dimension n≥2 satisfies the uniform unimodularity condition and the Thomsen stratification intersection-number condition, then any line bundle L on X with L·C≥n-1+p for every T-invariant curve C satisfies Property N_p. The two conditions are verified for several families and preserved under finite products.

What carries the argument

The uniform unimodularity condition combined with the Thomsen stratification intersection-number condition on the toric variety, which together allow the intersection bound on invariant curves to imply Property N_p for line bundles.

Load-bearing premise

The toric variety X must satisfy both the uniform unimodularity condition and the Thomsen stratification intersection-number condition.

What would settle it

A counterexample would be a smooth projective toric variety satisfying the two conditions together with a line bundle L where L·C ≥ n-1+p for all T-invariant curves but L fails Property N_p.

read the original abstract

We establish a criterion for Property $N_p$ for line bundles on a class of smooth projective toric varieties. More precisely, we prove that if a smooth projective toric variety $X$ of dimension $n\ge2$ satisfies the uniform unimodularity condition and the Thomsen stratification intersection-number condition, then any line bundle $L$ on $X$ with $L\cdot C\ge n-1+p$ for every $T$-invariant curve $C$ satisfies Property $N_p$. We also show that these two conditions hold for several families of toric varieties and are preserved under finite products.

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 manuscript proves that if a smooth projective toric variety X of dimension n≥2 satisfies the uniform unimodularity condition and the Thomsen stratification intersection-number condition, then any line bundle L on X with L·C ≥ n-1+p for every T-invariant curve C satisfies Property N_p. The authors additionally verify that these two conditions hold for several families of toric varieties and are preserved under finite products.

Significance. If the central implication holds, the result supplies an explicit numerical criterion for Property N_p on a nontrivial class of toric varieties, extending classical results known for projective space. The explicit verification of the two hypotheses on concrete families together with the product-stability statement increases the criterion's applicability and is a concrete strength of the work.

minor comments (2)
  1. The precise statement of the Thomsen stratification intersection-number condition (Definition 3.4) could be recalled verbatim in the introduction when the main theorem is first stated, to improve readability for readers who skip directly to §4.
  2. Notation for the uniform unimodularity condition is introduced in §2 but used without re-statement in the product-stability argument of §5; a one-sentence reminder would prevent the reader from paging back.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the clear summary of our main result, and the recommendation to accept. We are pleased that the explicit numerical criterion, the verifications on families, and the product-stability statement were viewed as strengths.

Circularity Check

0 steps flagged

No circularity: conditional criterion with independent hypotheses

full rationale

The central result is an implication: under the uniform unimodularity condition and the Thomsen stratification intersection-number condition (treated explicitly as extra hypotheses on X, verified separately for families and preserved under products), the intersection lower bound on L implies Property N_p. No equations, definitions, or self-citations in the abstract reduce the conclusion to a tautology or fitted input; the two conditions are not automatic and must be checked independently. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the two named conditions being well-defined and independent of the conclusion; these are domain assumptions in toric geometry rather than ad-hoc inventions. No free parameters or new entities are introduced in the abstract statement.

axioms (2)
  • standard math Smooth projective toric varieties are classified by fans in lattices satisfying standard combinatorial conditions.
    Invoked implicitly when speaking of T-invariant curves and line bundles on X.
  • standard math Property N_p is a well-defined notion from the theory of syzygies and Castelnuovo-Mumford regularity.
    The target property whose satisfaction is asserted.

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Reference graph

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