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arxiv: 2604.03471 · v1 · submitted 2026-04-03 · 🧮 math.AG · math.AC

On nice mathbb{G}_m-actions arising from locally nilpotent derivations with slice

Luis Cid This is my paper

Pith reviewed 2026-05-13 17:49 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords locally nilpotent derivationsG_m-actionslinearizabilityslicesautomorphic conjugacyaffine varieties
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The pith

In the nice case, a G_m-action from a locally nilpotent derivation with slice is linearizable exactly when the derivation is automorphically conjugate to partial over partial x_n and the slice is affine-linear in the distinguished variable.

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

The paper considers locally nilpotent derivations on finitely generated domains over algebraically closed fields of characteristic zero that admit a slice. Multiplying such a derivation by its slice produces a semisimple derivation that generates a regular G_m-action on the spectrum. In the nice case where the square of the derivation vanishes on every generator, the paper proves that the resulting G_m-action is linearizable if and only if the derivation is conjugate via an automorphism to the standard partial derivative in one variable and the slice is an affine-linear polynomial in that variable. The criterion holds independently of which slice is selected.

Core claim

Given a locally nilpotent derivation D that admits a slice s, the scaled derivation NsD is semisimple and defines a regular G_m-action. In the nice case, where D squared vanishes on all generators, this G_m-action is linearizable if and only if D is automorphically conjugate to partial over partial x_n and the slice becomes affine-linear in the distinguished variable; the criterion does not depend on the choice of slice.

What carries the argument

The linearizability criterion for nice G_m-actions, which requires automorphic conjugacy of the derivation to partial over partial x_n together with the slice being affine-linear in the distinguished variable.

If this is right

  • The G_m-action receives an explicit description via the semisimple infinitesimal generator NsD.
  • Linearizability becomes independent of the particular slice that is chosen.
  • The criterion applies directly to the G_m-action previously introduced by Freudenburg.
  • Linearizability reduces to checking automorphic conjugacy and affine-linearity of the slice.

Where Pith is reading between the lines

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

  • The slice-independence may simplify computational verification of linearizability for concrete low-dimensional examples.
  • The conjugacy condition could connect to broader questions about the structure of automorphism groups of polynomial rings.
  • The explicit generator description might extend to related questions on semisimple actions in higher dimensions.

Load-bearing premise

The derivation satisfies the nice condition that its square vanishes on each generator, and the ring is finitely generated over an algebraically closed field of characteristic zero.

What would settle it

A counterexample consisting of a nice locally nilpotent derivation with slice where the derivation is not automorphically conjugate to partial over partial x_n yet the associated G_m-action is still linearizable, or the converse situation.

read the original abstract

Let $k$ be an algebraically closed field of characteristic zero and $B$ a finitely generated $k$-domain. Given a locally nilpotent derivation $D$ on $B$ admitting a slice $s$, the derivation $\partial=NsD$ ($N\in\mathbb{Z}\setminus\{0\}$) is semisimple and defines a regular $\mathbb{G}_m$-action on $\mathrm{Spec}(B)$. We show that this derivation provides a new explicit description of the $\mathbb{G}_m$-action introduced by Freudenburg in terms of the infinitesimal generator $\partial=NsD$. In the nice case ($D^2(x_i)=0$ for all generators), we prove a linearizability criterion: the associated $\mathbb{G}_m$-action is linearizable if and only if $D$ is automorphically conjugate to $\partial/\partial x_n$ and the slice becomes affine-linear in the distinguished variable; moreover, this criterion is independent of the choice of slice.

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 / 3 minor

Summary. The paper studies locally nilpotent derivations D on a finitely generated k-domain B (k algebraically closed of characteristic zero) that admit a slice s. It defines the semisimple derivation ∂ = N s D (N nonzero integer) and shows that this provides an explicit infinitesimal generator for the regular G_m-action on Spec(B) introduced by Freudenburg. In the nice case (D²(x_i) = 0 for all generators x_i), it proves that the associated G_m-action is linearizable if and only if D is automorphically conjugate to ∂/∂x_n and the slice is affine-linear in the distinguished variable; the criterion is independent of the choice of slice.

Significance. If the central claims hold, the work supplies a concrete, slice-independent linearizability criterion for G_m-actions arising from nice LNDs with slices, together with an explicit description of the action via the generator ∂ = N s D. This strengthens the link between LND theory and linearizable actions on affine varieties and may facilitate explicit computations or classifications in algebraic geometry.

minor comments (3)
  1. Abstract: the phrase 'in the nice case' is used without a parenthetical reminder of the condition D²(x_i)=0; adding this improves immediate readability for readers scanning the abstract.
  2. Introduction: the reference to Freudenburg's original construction of the G_m-action should include a precise citation (paper or book section) so that the claimed 'new explicit description' can be compared directly.
  3. Notation section: the symbol N is introduced as a nonzero integer without stating whether the final linearizability statement is independent of the choice of N; a short remark would remove potential ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its contributions, and the recommendation for minor revision. We are pleased that the work is viewed as strengthening the link between LND theory and linearizable G_m-actions. Since no specific major comments were raised, we will incorporate any minor editorial or expository improvements in the revised version.

Circularity Check

0 steps flagged

No circularity: pure algebraic proof with independent derivations

full rationale

The paper is a self-contained mathematical derivation in algebraic geometry. It defines locally nilpotent derivations with slices, constructs the associated G_m-action via the explicit formula partial = N s D, and proves the linearizability criterion in the nice case (D^2(x_i)=0) by direct algebraic manipulation of generators and automorphisms. The criterion is shown equivalent to conjugacy to partial/partial x_n with affine-linear slice, using only the ring structure and standard properties of LNDs; no parameters are fitted to data, no central quantity is defined in terms of the result it claims to derive, and no load-bearing step reduces to a self-citation. The reference to Freudenburg's prior construction is external and used only for comparison, not as justification for uniqueness or the main theorem. All steps are verifiable from the given definitions and ring axioms without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from commutative algebra and algebraic geometry; no free parameters or invented entities are introduced.

axioms (2)
  • standard math k is an algebraically closed field of characteristic zero
    Invoked at the outset to guarantee the existence and properties of locally nilpotent derivations and G_m-actions.
  • domain assumption B is a finitely generated k-domain
    Required for Spec(B) to be an affine variety and for the derivation to act on a Noetherian ring.

pith-pipeline@v0.9.0 · 5469 in / 1324 out tokens · 69389 ms · 2026-05-13T17:49:58.863267+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    Cid and A

    L. Cid and A. Liendo,On rational multiplicative group actions, preprint, arXiv:2208.05024, 2022

    work page arXiv 2022

  2. [2]

    Cid,Semisimple derivations, rational slices, and kernels over affine domains, preprint, arXiv:2603.19587, 2026

    L. Cid,Semisimple derivations, rational slices, and kernels over affine domains, preprint, arXiv:2603.19587, 2026

    work page arXiv 2026

  3. [3]

    Freudenburg,Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences, Vol

    G. Freudenburg,Algebraic Theory of Locally Nilpotent Derivations, Encyclopaedia of Mathematical Sciences, Vol. 136, Springer-Verlag, Berlin, 2006

    work page 2006

  4. [4]

    Gutwirth,The action of an algebraic torus on the affine plane, Trans

    A. Gutwirth,The action of an algebraic torus on the affine plane, Trans. Amer. Math. Soc.105(1962), 407–414

    work page 1962

  5. [5]

    Kaliman, M

    S. Kaliman, M. Koras, L. Makar-Limanov, and P. Russell,C∗-actions onC 3 are linearizable, Electron. Res. Announc. Amer. Math. Soc.3(1997), 63–71

    work page 1997

  6. [6]

    Koras and P

    M. Koras and P. Russell,Contractible threefolds andC∗-actions onC 3, J. Algebraic Geom.6(1997), 671–695

    work page 1997

  7. [7]

    Wang,Homogenization of locally nilpotent derivations and an application tok[X, Y, Z], J

    Z. Wang,Homogenization of locally nilpotent derivations and an application tok[X, Y, Z], J. Pure Appl. Algebra196(2005), 323–337. Instituto de Matemática y Física, Universidad de Talca, Casilla 721, Talca, Chile Email address:luis.cid@inst-mat.utalca.cl

    work page 2005