REVIEW 2 minor
Chow groups and pseudoeffective cones of complexity one $T$-varieties
T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3
Pith's one-line read The pseudoeffective cone of k-cycles on a complete complexity one T-variety is rational polyhedral, generated by classes of T-invariant subvarieties.
desk verdict The paper gives a clean combinatorial description of pseudoeffective cones and Chow groups for complete complexity-one T-varieties using the defining data of the torus action. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The combinatorial data (fans or polytopes) of the complexity-one torus action, which determines the cycle classes of all T-invariant subvarieties.
What would settle it
A complete complexity-one T-variety together with an explicit k-cycle class that lies in the pseudoeffective cone but cannot be expressed as a nonnegative rational combination of classes of T-invariant subvarieties.
Extended reading notes
Core claim
We show that the pseudoeffective cone of k-cycles on a complete complexity one T-variety is rational polyhedral for any k, generated by classes of T-invariant subvarieties. When X is also rational, we give a presentation of the Chow groups of X in terms of generators and relations, coming from the combinatorial data defining X as a T-variety.
Load-bearing premise
The variety is complete and the complexity-one torus action has combinatorial data that fully determines the cycle classes of its T-invariant subvarieties.
Editorial extensions
If this is right
- The pseudoeffective cone of k-cycles is generated by T-invariant subvarieties for every k.
- Chow groups of rational complexity-one T-varieties are presented by generators and relations from the combinatorial data.
- Questions about effective cycles reduce to linear algebra over the classes of invariant subvarieties.
Reading between the lines
- The same combinatorial generators may allow explicit computation of the movable cone or other cones of cycles.
- The result supplies a test case for whether similar polyhedrality holds for T-varieties of higher complexity under additional hypotheses.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper shows that the pseudoeffective cone of k-cycles on a complete complexity one T-variety is rational polyhedral for any k, generated by classes of T-invariant subvarieties. When X is also rational, it gives a presentation of the Chow groups of X in terms of generators and relations coming from the combinatorial data defining X as a T-variety.
Significance. If the result holds, it extends known combinatorial descriptions of pseudoeffective cones and Chow groups from toric varieties to the larger class of complexity-one T-varieties. The explicit combinatorial proof using the polyhedral data of the T-action and the completeness assumption is a strength, as it supplies a concrete, verifiable method for determining these objects directly from the defining fans or polytopes.
minor comments (2)
- The abstract could briefly indicate the dimension range or provide a low-dimensional example to illustrate the generators-and-relations presentation for Chow groups.
- Notation for the combinatorial data (fans, polytopes) should be introduced with a short reminder in the introduction for readers unfamiliar with T-variety literature.
Simulated Author's Rebuttal
We thank the referee for the positive report, the accurate summary of our results, and the recommendation to accept. No major comments require a point-by-point response.
Circularity Check
No significant circularity detected
full rationale
The paper derives the rational polyhedrality of the pseudoeffective cone of k-cycles on complete complexity-one T-varieties directly from the combinatorial data (fans/polytopes) of the torus action together with the completeness assumption, presenting an explicit combinatorial proof that the cone is generated by classes of T-invariant subvarieties. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central claims remain independent of the result itself and are not forced by definition or prior author work.
Assumptions & free parameters
assumptions (1)
- standard math Standard properties of Chow groups, rational equivalence, and pseudoeffective cones on complete varieties
Cite this review
Pith. "Pith review of Chow groups and pseudoeffective cones of complexity one $T$-varieties." pith.science (2026). https://pith.science/paper/ZEFVGIPU
@misc{pith2026190710941,
author = {Pith},
title = {Pith review of: Chow groups and pseudoeffective cones of complexity one $T$-varieties},
year = {2026},
howpublished = {\url{https://pith.science/paper/ZEFVGIPU}},
note = {Machine review of arXiv:1907.10941}
}
abstract
We show that the pseudoeffective cone of $k$-cycles on a complete complexity one $T$-variety is rational polyhedral for any $k$, generated by classes of $T$-invariant subvarieties. When $X$ is also rational, we give a presentation of the Chow groups of $X$ in terms of generators and relations, coming from the combinatorial data defining $X$ as a $T$-variety.
Reviewed May 24, 2026 · model on record in the stance chip above.
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