Cylinders and the zero locus of the plinth ideal

Kirill Shakhmatov

arxiv: 2605.00138 · v1 · submitted 2026-04-30 · 🧮 math.AG

Cylinders and the zero locus of the plinth ideal

Kirill Shakhmatov This is my paper

Pith reviewed 2026-05-09 20:33 UTC · model grok-4.3

classification 🧮 math.AG

keywords Ga-actionaffine varietyplinth ideallocally nilpotent derivationinvariant cylinderzero locusalgebraic group action

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The pith

For a Ga-action on an affine variety X, the complement of the union of all principal invariant cylinders equals the zero locus of the plinth ideal.

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

The paper proves that in an affine variety equipped with an action of the additive group Ga, the points that avoid every principal invariant cylinder coincide exactly with the common zeros of the plinth ideal coming from the locally nilpotent derivation that defines the action. This links a geometric covering by invariant lines or curves to a purely algebraic ideal. A reader would care because the equality turns the problem of locating free orbits or fixed loci into an ideal-membership question that can be checked on the coordinate ring.

Core claim

Given a Ga-action on an affine variety X, we show that the complement of the union of all principal invariant cylinders in X is equal to the zero locus of the plinth ideal of the corresponding locally nilpotent derivation.

What carries the argument

The plinth ideal of the locally nilpotent derivation, whose zero set captures exactly the points missed by all principal invariant cylinders.

If this is right

The zero locus of the plinth ideal determines the regions through which no principal cylinder passes.
Existence of a principal cylinder through a given point can be decided by checking membership in the plinth ideal.
Algebraic generators of the plinth ideal give explicit equations cutting out the non-cylinder locus.

Where Pith is reading between the lines

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

The same equality could be used to classify Ga-actions whose plinth ideal is principal or generated in low degree.
On specific varieties such as affine space or toric varieties the result would give explicit equations for the non-cylinder set.
It may be possible to extend the statement to other algebraic group actions that admit analogous cylinder decompositions.

Load-bearing premise

The Ga-action arises from a locally nilpotent derivation on the coordinate ring of X and the principal invariant cylinders are those associated to the derivation in the usual way.

What would settle it

Exhibit a concrete affine variety X with a Ga-action where some point lies outside every principal invariant cylinder yet does not vanish on the plinth ideal, or the reverse.

read the original abstract

Given a $\mathbb{G}_\mathrm{a}$-action on an affine variety $X$, we show that the complement of the union of all principal invariant cylinders in $X$ is equal to the zero locus of the plinth ideal of the corresponding locally nilpotent derivation.

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

This short note records a direct equality between the complement of principal invariant cylinders and the zero locus of the plinth ideal, which follows immediately from the standard LND-Ga correspondence.

read the letter

The paper states that for a Ga-action on an affine variety X the points outside every principal invariant cylinder are exactly the zeros of the plinth ideal of the associated locally nilpotent derivation. It proves both inclusions by invoking the slice criterion for the existence of a local slice and the fact that D(f) generates the plinth ideal when f is invariant and a local slice exists. The argument uses only the usual bijection between Ga-actions in characteristic zero and LNDs on the coordinate ring, with no extra hypotheses on smoothness or freeness of the action. That is the main content. The write-up is clean and stays inside the standard definitions, so the claim checks out once the definitions of principal cylinder and plinth ideal are fixed. The only real limitation is that both directions rest on facts already in the literature on LNDs and invariant open sets, so the result is essentially an observation rather than a substantial new construction. Still, spelling the equality out explicitly can save time for people who work regularly with cylinders and plinth ideals. The paper is aimed at readers already inside affine algebraic geometry who care about Ga-actions and automorphism groups. It is not the sort of thing that changes the direction of the field, but the statement is precise enough that a short note like this can be useful as a reference. I would send it to a referee. The proof is short, the assumptions are standard, and the claim is easy to verify or falsify from the text.

Referee Report

0 major / 1 minor

Summary. The paper proves that for a Ga-action on an affine variety X induced by a locally nilpotent derivation D, the complement of the union of all principal invariant cylinders equals the zero locus of the plinth ideal pl(D) of D.

Significance. If the result holds, it gives a direct geometric interpretation of the zero locus of the plinth ideal as the locus where no principal invariant cylinder exists. The argument relies only on the standard bijection between Ga-actions and LNDs together with the slice criterion, without extra hypotheses on smoothness, reductivity, or freeness; this is a clean, parameter-free clarification within the existing theory of algebraic Ga-actions.

minor comments (1)

The introduction could briefly recall the definition of a principal invariant cylinder (as an open set X_f admitting a slice) to make the statement self-contained for readers less familiar with the slice criterion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary accurately captures the main result, and we appreciate the recognition that the argument relies only on the standard correspondence between Ga-actions and locally nilpotent derivations together with the slice criterion.

Circularity Check

0 steps flagged

No significant circularity; equality follows directly from definitions and slice criterion

full rationale

The central claim equates the complement of the union of principal invariant cylinders to the zero locus of the plinth ideal. This is established by proving both set inclusions directly: one direction uses the slice criterion to show that points outside the plinth zero locus admit an invariant open set X_f with a slice, while the converse uses that D(f) generates the plinth ideal for any local slice f. The argument invokes only the standard bijection between Ga-actions on affine varieties (char 0) and locally nilpotent derivations, together with the definitions of cylinders and the plinth ideal pl(D) as the ideal in ker(D) generated by D(A). No step reduces to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation; the derivation is self-contained against external algebraic geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard definitions and properties of Ga-actions, LNDs, and plinth ideals from prior literature in algebraic geometry; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Ga-actions on affine varieties correspond to locally nilpotent derivations on the coordinate ring
Standard correspondence used throughout the field of affine algebraic geometry.
domain assumption Principal invariant cylinders are defined via the LND in the usual way
Relies on established definitions of cylinders for LNDs.

pith-pipeline@v0.9.0 · 5319 in / 1156 out tokens · 38838 ms · 2026-05-09T20:33:55.569011+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg. Group actions on affine cones. CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, RI, 2011, 123–163

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Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg.G a-actions on affine cones. Transform. Groups 18 (2013), no. 4, 1137–1153

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Shakhmatov and H

Kirill Shakhmatov and Hoang Le Truong. Flexibility of affine cones over a smooth complete intersection of two quadrics. arXiv 2512.05219 (2025), 15 pp. To appear in Proc. Roy. Soc. Edinburgh Sect. A HSE University, F aculty of Computer Science, Pokrovsky Boulevard 11, Moscow, 109028 Russia Email address:kshahmatov@hse.ru

work page arXiv 2025

[1] [1]

Affine cones over smooth cubic surfaces

Ivan Cheltsov, Jihun Park, and Joonyeong Won. Affine cones over smooth cubic surfaces. J. Eur. Math. Soc. 18 (2016), no. 7, 1537–1564

work page 2016

[2] [2]

A note on locally nilpotent derivations and variables ofk[X, Y, Z]

Daniel Daigle and Shulim Kaliman. A note on locally nilpotent derivations and variables ofk[X, Y, Z]. Canad. Math. Bull. 52 (2009), no. 4, 535–543

work page 2009

[3] [3]

Springer, Heidelberg, 2017

Gene Freudenburg.Algebraic Theory of Locally Nilpotent Derivations. Springer, Heidelberg, 2017

work page 2017

[4] [4]

Automorphisms of Danielewski varieties

Sergey Gaifullin. Automorphisms of Danielewski varieties. J. Algebra 573 (2021), 364–392

work page 2021

[5] [5]

On the cancellation problem for the affine spaceA 3 in characteristicp

Neena Gupta. On the cancellation problem for the affine spaceA 3 in characteristicp. Invent. Math. 195 (2014), no. 1, 279–288

work page 2014

[6] [6]

Group actions on affine cones

Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg. Group actions on affine cones. CRM Proc. Lecture Notes 54, Amer. Math. Soc., Providence, RI, 2011, 123–163

work page 2011

[7] [7]

Transform

Takashi Kishimoto, Yuri Prokhorov, and Mikhail Zaidenberg.G a-actions on affine cones. Transform. Groups 18 (2013), no. 4, 1137–1153

work page 2013

[8] [8]

On the group of automorphisms of a surfacex ny=P(z)

Leonid Makar-Limanov. On the group of automorphisms of a surfacex ny=P(z). Israel J. Math. 121 (2001), 113–123

work page 2001

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Factorial affineG a-varieties with height one plinth ideals

Kayo Masuda. Factorial affineG a-varieties with height one plinth ideals. Transform. Groups 30 (2025), no. 4, 1887–1914

work page 2025

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Op´ erations du groupe additif sur le plan affine

Rudolf Rentschler. Op´ erations du groupe additif sur le plan affine. C. R. Acad. Sci. Paris S´ er. A-B 267 (1968), 384–387 8 KIRILL SHAKHMATOV

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Shakhmatov and H

Kirill Shakhmatov and Hoang Le Truong. Flexibility of affine cones over a smooth complete intersection of two quadrics. arXiv 2512.05219 (2025), 15 pp. To appear in Proc. Roy. Soc. Edinburgh Sect. A HSE University, F aculty of Computer Science, Pokrovsky Boulevard 11, Moscow, 109028 Russia Email address:kshahmatov@hse.ru

work page arXiv 2025