Chow groups and pseudoeffective cones of complexity one $T$-varieties

Bernt Ivar Utst{\o}l N{\o}dland

arxiv: 1907.10941 · v1 · pith:ZEFVGIPUnew · submitted 2019-07-25 · 🧮 math.AG

Chow groups and pseudoeffective cones of complexity one T-varieties

Bernt Ivar Utst{\o}l N{\o}dland This is my paper

Pith reviewed 2026-05-24 16:22 UTC · model grok-4.3

classification 🧮 math.AG

keywords pseudoeffective coneChow groupsT-varietiescomplexity onetorus actioninvariant subvarietiesrational polyhedralcycle classes

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The pith

The pseudoeffective cone of k-cycles on a complete complexity one T-variety is rational polyhedral, generated by classes of T-invariant subvarieties.

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

The paper shows that the pseudoeffective cone of k-cycles on any complete complexity one T-variety is a rational polyhedral cone generated by the classes of T-invariant subvarieties, for every dimension k. When the variety is additionally rational, the Chow groups admit an explicit presentation by generators and relations coming directly from the combinatorial data of the torus action. A sympathetic reader would care because this turns questions about which cycle classes are effective into questions about nonnegative linear combinations of invariant subvarieties whose classes are known combinatorially.

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.

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.

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.

Where Pith is reading between the lines

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

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.

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.

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.

read the original 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.

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.

Desk Editor's Note private letter to a colleague

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 main takeaway is that for any complete complexity-one T-variety the pseudoeffective cone of k-cycles is rational polyhedral and generated by classes of T-invariant subvarieties, with an explicit presentation of the Chow groups when the variety is also rational. Both statements are derived directly from the combinatorial data that defines the T-action. This is new: earlier work on T-varieties handled lower-dimensional cases or toric varieties, but the explicit polyhedrality and generator description for arbitrary k in the complexity-one setting had not been recorded. The paper does well by turning the geometric questions into polyhedral combinatorics once the fan or polytope data is fixed, which is the natural language for these varieties. The argument stays within that language and uses completeness to control the cones. The soft spots are modest and expected. Everything rests on the variety being complete and the action having complexity exactly one; drop either condition and the statements need separate justification. The proof is presented as combinatorial, so verification reduces to checking the generators and relations against the given data rather than to new geometric input. No circularity or hidden non-combinatorial cycles appear in the outline. This paper is for people who already work with T-varieties or with effective cones on varieties carrying group actions. A reader comfortable with toric geometry and the basic dictionary between invariant subvarieties and polyhedral data will extract the most value. It is narrow in scope but the claims are concrete and the method is direct, so it deserves a serious referee who can check the combinatorial steps in detail.

Referee Report

0 major / 2 minor

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

0 responses · 0 unresolved

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

0 steps flagged

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard axioms of algebraic geometry (Chow groups, pseudoeffective cones) together with the definition of complete complexity-one T-varieties; no free parameters or new entities are introduced.

axioms (1)

standard math Standard properties of Chow groups, rational equivalence, and pseudoeffective cones on complete varieties
The paper invokes the usual functoriality and positivity properties of cycle classes in algebraic geometry.

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