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The Mori cone of certain Hassett spaces

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Hassett spaces whose universal family is a P¹-bundle have Mori cones generated by 1-dimensional strata.

desk verdict The paper proves that the Mori cone of Hassett spaces with P¹-bundle universal family is generated by 1-dimensional strata, extending the Bolognesi-Massarenti GIT case via reviewed isomorphisms and contraction characterizations. read the letter →

arxiv 2606.22012 v1 pith:4RI7SOSZ submitted 2026-06-20 math.AG

Davide Ricci This is my paper · ORCID
classification math.AG
keywords HassettspacesMoriconeGITquotientseffectiveP¹-bundlebirationalcontractionsblow-upofprojectivespacestrata
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the Mori cone of these specific Hassett spaces is generated by 1-dimensional strata. It extends the known case for the symmetric GIT quotient of (P¹)^n by PGL₂. The spaces are shown to be isomorphic to certain GIT quotients and to arise as targets of birational contractions from the blow-up of P^{n-3} at n-1 general points, with Q-factorial image. As a consequence, their effective cone is also generated by strata. A reader cares because this describes the cone of curves on these moduli spaces in terms of explicit geometric generators.

What carries the argument

The 1-dimensional strata, which generate the Mori cone (and effective cone) on these Hassett spaces via their identification with GIT quotients and contraction targets.

What would settle it

For a concrete n greater than 5, compute the Mori cone of the corresponding Hassett space by finding a curve class not in the cone spanned by the 1-dimensional strata and check whether it lies in the Mori cone.

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Extended reading notes

Core claim

We prove that the Mori cone of Hassett spaces whose universal family is a P¹-bundle is generated by 1-dimensional strata. This extends the case of the symmetric GIT quotient (P¹)^n//PGL₂. Along the way, we review how these spaces are naturally isomorphic to certain GIT quotients (P¹)^n//PGL₂, characterize them as the targets of the birational contractions of the blow-up of P^{n-3} at n-1 general points with Q-factorial image, and deduce that their effective cone is likewise generated by strata.

Load-bearing premise

The spaces under consideration are precisely the targets of the birational contractions of the blow-up of P^{n-3} at n-1 general points that have Q-factorial image.

Editorial extensions

If this is right

  • The effective cone of these Hassett spaces is generated by strata.
  • These Hassett spaces are isomorphic to specific GIT quotients (P¹)^n//PGL₂.
  • The birational contractions from the blow-up of P^{n-3} at n-1 points land on spaces with Q-factorial image whose cones are stratum-generated.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The result may extend to other classes of Hassett spaces beyond those with P¹-bundle universal families.
  • Similar generation statements could hold for the Mori cones of related moduli spaces obtained by different contractions.
  • Explicit generators for the cones might allow computation of the Picard rank or other invariants in these cases.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper proves that the Mori cone of Hassett spaces whose universal family is a P¹-bundle is generated by 1-dimensional strata. This extends the Bolognesi-Massarenti result for the symmetric GIT quotient (P¹)^n // PGL₂. The spaces are shown to be isomorphic to certain GIT quotients (P¹)^n // PGL₂, characterized as targets of birational contractions of the blow-up of P^{n-3} at n-1 general points with Q-factorial image, and the effective cone is deduced to be generated by strata.

Significance. If the result holds, it completes the description of the Mori cone for this family of Hassett spaces by identifying explicit generators, extends a prior theorem to a broader class via standard GIT identifications, and connects the geometry to blow-ups of projective space. The additional characterization of the spaces and the effective-cone statement are natural corollaries that strengthen the contribution to the birational geometry of moduli spaces.

minor comments (2)
  1. The abstract states that the spaces are 'naturally isomorphic' to GIT quotients; a brief sentence in the introduction recalling the precise weight vector or stability condition used for the isomorphism would help readers who are not specialists in Hassett spaces.
  2. Notation for the 1-dimensional strata (e.g., whether they are denoted by boundary divisors or by specific curve classes) is introduced without an explicit cross-reference to the section where the generators are listed; adding a forward reference would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report accurately summarizes the main results, including the extension of the Bolognesi-Massarenti theorem and the additional characterizations provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper proves its central claim by extending the independent Bolognesi-Massarenti theorem on the symmetric GIT quotient (P¹)^n//PGL₂ (different authors) and by reviewing standard isomorphisms to GIT quotients plus the characterization of the spaces as targets of birational contractions from the blow-up of P^{n-3} at n-1 points. These steps are presented as theorems with external support rather than self-referential definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against external benchmarks and does not reduce any result to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based solely on the abstract; no explicit free parameters, new axioms, or invented entities are stated. The work relies on standard background in algebraic geometry and moduli theory.

assumptions (1)
  • standard math Standard results on Mori cones and effective cones of moduli spaces of curves
    Invoked implicitly when stating that the cone is generated by strata.

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Cite this review

Pith. "Pith review of The Mori cone of certain Hassett spaces." pith.science (2026). https://pith.science/paper/4RI7SOSZ

@misc{pith2026260622012,
  author       = {Pith},
  title        = {Pith review of: The Mori cone of certain Hassett spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4RI7SOSZ}},
  note         = {Machine review of arXiv:2606.22012}
}
abstract

In this paper, we prove that the Mori cone of Hassett spaces whose universal family is a $\mathbf{P}^1$-bundle is generated by 1-dimensional strata. This extends the case of the symmetric GIT quotient $(\mathbf{P}^1)^n//PGL_2$ established by Bolognesi and Massarenti. Along the way, we review how these spaces are naturally isomorphic to certain GIT quotients $(\mathbf{P}^1)^n//PGL_2$, characterize them as the targets of the birational contractions of the blow-up of $\mathbf{P}^{n-3}$ at $n-1$ general points with $\mathbf{Q}$-factorial image, and deduce that their effective cone is likewise generated by strata.

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Reference graph

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Reviewed June 26, 2026 · model on record in the stance chip above.