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arxiv: 2411.17370 · v3 · pith:323LO3PBnew · submitted 2024-11-26 · 🧮 math.AG

The Cox ring of an embedded variety

Crist\'obal Herrera , Antonio Laface , Luca Ugaglia This is my paper

Pith reviewed 2026-05-23 16:38 UTC · model grok-4.3

classification 🧮 math.AG
keywords Cox ringMori dream spaceembedded varietydivisor class grouptoric varietyfinite generation
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The pith

The Cox ring of an embedded variety X in a Mori dream space Z is the intersection of finitely many localizations of a quotient image of the Cox ring of Z when pullback induces an isomorphism on divisor class groups.

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

The paper shows how to obtain the Cox ring of a subvariety X inside a Mori dream space Z from the Cox ring of Z under the assumption that the pullback map gives an isomorphism on divisor class groups. It proves that the Cox ring of X arises as the intersection of finitely many localizations of a quotient image of the Cox ring of Z. This description produces a terminating algorithm that decides finite generation of the Cox ring of X. The method is applied to hypersurfaces inside smooth projective toric varieties and extends earlier explicit computations.

Core claim

If X is embedded in a Mori dream space Z and the pullback map induces an isomorphism on divisor class groups, then the Cox ring of X equals the intersection of finitely many localizations of a quotient image of the Cox ring of Z. This relation yields an algorithm that terminates if and only if the Cox ring of X is finitely generated.

What carries the argument

The central construction is the expression of the Cox ring of X as the intersection of finitely many localizations of a quotient image of the Cox ring of Z.

If this is right

  • Finite generation of the Cox ring of X is decidable by running the localization-and-intersection algorithm and checking termination.
  • The Cox rings of hypersurfaces in smooth projective toric varieties can be computed explicitly from the ambient toric Cox ring.
  • Earlier explicit computations of Cox rings for embedded varieties are recovered and extended by the same intersection procedure.

Where Pith is reading between the lines

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

  • The same localization-intersection description may apply to other embeddings where the class group condition holds, even outside toric ambient spaces.
  • If the algorithm terminates for a given X, the resulting ring presentation could be used to study the effective cone or movable cone of X directly.

Load-bearing premise

The pullback map induces an isomorphism at the level of divisor class groups.

What would settle it

A concrete hypersurface in a smooth projective toric variety where the intersection of localizations of the ambient Cox ring quotient differs from the directly computed Cox ring of the hypersurface.

read the original abstract

We compute the Cox ring of an embedded variety $X \subseteq Z$ within a Mori dream space, under the assumption that the pullback map induces an isomorphism at the level of divisor class groups. We show that the Cox ring of $X$ is the intersection of finitely many localizations of a quotient image of the Cox ring of $Z$. As a consequence, we provide an algorithm that terminates if and only if the Cox ring of $X$ is finitely generated, thereby generalizing previous works on the subject. We apply these results to compute the Cox ring of hypersurfaces in smooth projective toric varieties.

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 claims that for an embedded variety X ⊆ Z with Z a Mori dream space, assuming the pullback induces an isomorphism Cl(Z) ≅ Cl(X), the Cox ring of X equals the intersection of finitely many localizations of a quotient image of the Cox ring of Z. This yields an algorithm terminating if and only if Cox(X) is finitely generated, generalizing prior work, with applications to hypersurfaces in smooth projective toric varieties.

Significance. If the result holds, the explicit localization-based description and the associated termination-detecting algorithm constitute a useful computational tool for Cox rings in embedded settings. The generalization of earlier algorithms on Mori dream spaces, together with the concrete toric hypersurface applications, strengthens the practical reach of Cox ring computations in algebraic geometry.

minor comments (2)
  1. The precise construction of the 'quotient image' and the choice of the finitely many localizations should be stated with explicit reference to the irrelevant ideal and the grading in the main theorem (likely Theorem 3.1 or equivalent).
  2. In the application section on toric hypersurfaces, verify and state explicitly for which degrees or classes the Cl-isomorphism hypothesis holds, rather than leaving it implicit.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript, recognition of its significance as a computational tool, and recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained algebraic construction

full rationale

The central result states that, under the explicit hypothesis that the pullback induces Cl(Z) ≅ Cl(X), the Cox ring of X equals the intersection of finitely many localizations of a quotient image of the Cox ring of Z. This is presented as a direct algebraic description (standard in the Mori dream space literature) rather than a fitted parameter, self-definition, or output forced by prior self-citation. The algorithm for detecting finite generation follows immediately from the description and is not circular. No load-bearing self-citation, uniqueness theorem imported from the authors, or renaming of known results appears in the given material; the claim remains independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the single stated assumption that the pullback induces an isomorphism on divisor class groups; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption pullback map induces an isomorphism at the level of divisor class groups
    Explicitly listed in the abstract as the hypothesis under which the intersection description holds.

pith-pipeline@v0.9.0 · 5622 in / 1215 out tokens · 22547 ms · 2026-05-23T16:38:29.829987+00:00 · methodology

discussion (0)

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Reference graph

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