Infinitesimal rational actions

Bianca Gouthier

arxiv: 2312.01765 · v3 · pith:KWGC33IMnew · submitted 2023-12-04 · 🧮 math.AG

Infinitesimal rational actions

Bianca Gouthier This is my paper

Pith reviewed 2026-05-24 04:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords group schemesrational actionsinfinitesimal groupstrigonalizable groupspositive characteristicgenerically free actionsLie algebra dimensionalgebraic geometry

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The pith

Lie algebra dimension is the sole obstruction to generically free rational actions of infinitesimal commutative trigonalizable group schemes over perfect fields of positive characteristic.

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

For any finite group scheme acting rationally and generically freely on a variety, the dimension of its Lie algebra cannot exceed the dimension of the variety. The paper proves this bound is also sufficient for existence of such actions when the base field is perfect of positive characteristic and the group is infinitesimal, commutative, and trigonalizable. It further supplies necessary conditions for faithful actions in this setting and separate sufficient conditions when the group is unipotent. A sympathetic reader sees this as reducing the existence question to a single numerical comparison under the stated hypotheses.

Core claim

We show that the dimension of Lie(G) being at most the dimension of the variety is the only obstruction to the existence of a generically free rational action when k is a perfect field of positive characteristic and G is an infinitesimal commutative trigonalizable group scheme. We also give necessary conditions to have faithful rational actions of such group schemes on varieties, and sufficient conditions in the unipotent case over a perfect field.

What carries the argument

The Lie algebra dimension of an infinitesimal commutative trigonalizable group scheme G, which controls existence of generically free rational actions on a variety precisely when it does not exceed the variety dimension.

If this is right

If dim Lie(G) exceeds dim X then no generically free rational G-action on X exists, for any finite k-group scheme.
When the dimension inequality holds and the structural hypotheses on G and k are met, a variety X of that dimension carrying a generically free rational G-action must exist.
Faithful rational actions of these group schemes require additional necessary conditions beyond the dimension bound.
In the unipotent case over a perfect field, separate sufficient conditions guarantee the existence of the actions.

Where Pith is reading between the lines

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

The result isolates the exact hypotheses under which dimension alone decides existence, suggesting that relaxing commutativity or infinitesimalness would require new invariants.
Explicit constructions for low-dimensional examples could test the boundary between the necessary and sufficient conditions given.
Links to the broader classification of finite group schemes in positive characteristic may clarify when the trigonalizable hypothesis can be weakened.

Load-bearing premise

The group scheme must be infinitesimal, commutative, and trigonalizable over a perfect field of positive characteristic.

What would settle it

An explicit infinitesimal commutative trigonalizable group scheme G over a perfect field of positive characteristic with dim Lie(G) at most n, together with a proof that no variety of dimension n admits a generically free rational G-action.

read the original abstract

For any finite $k$-group scheme $G$ acting rationally on a $k$-variety, if the action is generically free then the dimension of $\mathrm{Lie} (G)$ is upper bounded by the dimension of the variety. We show that this is the only obstruction when $k$ is a perfect field of positive characteristic and $G$ is infinitesimal commutative trigonalizable. We also give necessary conditions to have faithful rational actions of infinitesimal commutative trigonalizable group schemes on varieties, and (different) sufficient conditions in the unipotent case over a perfect field.

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 proves that a known dimension bound on Lie(G) is the only obstruction to generically free rational actions when G is infinitesimal commutative trigonalizable over a perfect positive-characteristic field.

read the letter

The key point here is that the paper establishes sufficiency for the dimension bound being the only obstruction to generically free rational actions, but only when the field is perfect of positive characteristic and the group scheme G is infinitesimal, commutative, and trigonalizable. This turns a known necessary condition into a sufficient one under those restrictions. The work also provides necessary conditions for faithful rational actions of such group schemes and gives different sufficient conditions in the unipotent case over a perfect field. It stays focused on finite k-group schemes acting rationally on k-varieties. This is new in the sense that it resolves the obstruction question in this restricted class, which the abstract presents as extending prior necessary conditions. The paper does well by being explicit about the hypotheses and not claiming more generality than it has. The central theorem is stated clearly. The main limitation is the narrow scope. The sufficiency does not hold without the structural hypotheses on G and the field, as noted in the reader's weakest assumption. If those are dropped, the bound may not suffice. Since the full text was not reviewed here but the stress test found no internal inconsistency, the math appears to support the scoped claim. This paper is aimed at researchers in algebraic geometry working with positive-characteristic fields and infinitesimal group schemes. Someone in that subfield would find the conditions and the unipotent case useful for their own work on actions. It is not broad enough for general interest. I would not bring this to a general reading group, but maybe to one focused on algebraic groups. I would not cite it in my own work in the next year. The paper shows clear thinking and honest engagement with the literature on its terms. It deserves to go through peer review so the proofs can be checked in detail.

Referee Report

0 major / 2 minor

Summary. The paper proves that for any finite k-group scheme G acting rationally on a k-variety X, a generically free action implies dim Lie(G) ≤ dim X. It shows that this dimension bound is the only obstruction to existence of such actions when k is perfect of positive characteristic and G is infinitesimal, commutative, and trigonalizable. It also provides necessary conditions for faithful rational actions of such G and (separate) sufficient conditions in the unipotent case over a perfect field.

Significance. If the proofs hold, the result gives a sharp characterization of the obstruction to generically free rational actions for this class of group schemes, which is a useful contribution to the theory of infinitesimal group actions in positive characteristic. The explicit scoping of the sufficiency statement to the stated hypotheses on k and G is a strength, as is the separation of the faithful-action results from the generically-free case.

minor comments (2)

[Abstract] Abstract: the central theorem is stated without any indication of the proof strategy or key technical tools (e.g., whether the argument proceeds via explicit constructions, deformation theory, or classification of trigonalizable groups); adding one sentence would improve readability without lengthening the abstract unduly.
The manuscript should include a short paragraph in the introduction clarifying the precise meaning of 'trigonalizable' and 'infinitesimal' in this context, with a reference to a standard source if the definitions are not self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated hypotheses

full rationale

The paper states a dimension bound on Lie(G) for generically free actions and proves sufficiency only when k is perfect of positive characteristic and G is infinitesimal, commutative, and trigonalizable. No equations or claims reduce a prediction to a fitted input by construction, no load-bearing self-citations are invoked to justify uniqueness or ansatzes, and the result is explicitly scoped without renaming known results or smuggling assumptions. The central claim rests on direct proof within the given structural hypotheses rather than circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Central claim rests on standard definitions of rational actions, generically free actions, Lie algebras of group schemes, and the notions of infinitesimal, commutative, trigonalizable, and unipotent group schemes; no free parameters or invented entities appear in the abstract.

axioms (1)

standard math Standard properties of group schemes, their Lie algebras, and rational actions on varieties over a field (as in standard references on algebraic groups).
The statements presuppose the usual framework of algebraic geometry in positive characteristic.

pith-pipeline@v0.9.0 · 5606 in / 1172 out tokens · 28505 ms · 2026-05-24T04:58:44.359445+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

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Which abelian surfaces are products of elliptic curves?

[Oor75] Frans Oort. “Which abelian surfaces are products of elliptic curves?” In: Math. Ann. 214 (1975), pp. 35–47. [SGA3] Michael Artin, Jean-Etienne Bertin, Michel Demazure , Alexander Grothendieck, Pierre Gabriel, Michel Raynaud, and Jean-Pierre Serre. Schémas en groupes . Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientiﬁques. ...

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[Tos19] Dajano Tossici. “Essential dimension of inﬁnitesi mal commutative unipotent group schemes”. In: Boll. Unione Mat. Ital. 12.4 (2019), pp. 575–581. [TV13] Dajano Tossici and Angelo Vistoli. “On the essential dimension of inﬁnitesimal group schemes”. In: Amer. J. Math. 135.1 (2013), pp. 103–114. Institut de Mathématiques de Bordeaux, 351 Cours de la ...

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[1] [1]

Schéma d’automorphisms. Applicati on à l’étude d’extensions ﬁnies radicielles

[Beg69] Lucile Begueri. “Schéma d’automorphisms. Applicati on à l’étude d’extensions ﬁnies radicielles”. In: Bull. Sci. Math. (2) 93 (1969), pp. 89–111. [BF03] Grégory Berhuy and Giordano Favi. “Essential dimensio n: a functorial point of view (after A. Merkurjev)”. In: Doc. Math. 8 (2003), pp. 279–330. [BM11] Sylvain Brochard and Ariane Mézard. “About de...

work page 1969

[2] [2]

On the essential dimensio n of a ﬁnite group

[BR97] J. Buhler and Z. Reichstein. “On the essential dimensio n of a ﬁnite group”. In: Compositio Math. 106.2 (1997), pp. 159–179. [Bri22] Michel Brion. Actions of ﬁnite group schemes on curves

work page 1997

[3] [3]

[Bri22b] , On models of algebraic group actions , Proc

arXiv: 2207.08209 [math.AG] . [Cha72] Stephen U. Chase. “On the automorphism scheme of a pu rely inseparable ﬁeld extension”. In: Ring theory (Proc. Conf., Park City, Utah,

work page arXiv

[4] [4]

On elements of order ps in the plane Cremona group over a ﬁeld of characteristic p

Academic Press, New York-London, 1972, pp. 75–106. [DG70] Michel Demazure and Pierre Gabriel. Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. A vec un appendiceCorps de classes local par Michiel Hazewinkel. Masson & Cie, Éditeurs, Paris and No rth-Holland Publishing Co., Amsterdam, 1970, pp. xxvi+700. 40 [Dol09] I. V. ...

work page 1972

[5] [5]

Finite group schemes of ess ential dimension one

Adv. Stud. Pure Math. Math. Soc. Japan, Tokyo, 2010, pp. 1–49. [Fak20] Najmuddin Fakhruddin. “Finite group schemes of ess ential dimension one”. In: Doc. Math. 25 (2020), pp. 55–64. [Gou] Bianca Gouthier. Inﬁnitesimal commutative group schemes with one-dimensio nal Lie algebra. In preparation. [Liu02] Qing Liu. Algebraic geometry and arithmetic curves . Vol

work page 2010

[6] [6]

Representabili ty of group functors, and automorphisms of algebraic schemes

[MO67] Hideyuki Matsumura and Frans Oort. “Representabili ty of group functors, and automorphisms of algebraic schemes”. In: Invent. Math. 4 (1967), pp. 1–25. [Mon93] Susan Montgomery. Hopf algebras and their actions on rings . Vol

work page 1967

[7] [7]

Published for t he Conference Board of the Mathematical Sciences, Washington, DC; by the Americ an Mathematical Society, Providence, RI, 1993, pp

CBMS Regional Conference Series in Mathematics. Published for t he Conference Board of the Mathematical Sciences, Washington, DC; by the Americ an Mathematical Society, Providence, RI, 1993, pp. xiv+238. [Mum08] David Mumford. Abelian varieties . Vol

work page 1993

[8] [8]

With appendices by C

Tata Institute of Fundamental Re- search Studies in Mathematics. With appendices by C. P. Rama nujam and Yuri Manin, Corrected reprint of the second (1974) edition. Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New D elhi,

work page 1974

[9] [9]

Which abelian surfaces are products of elliptic curves?

[Oor75] Frans Oort. “Which abelian surfaces are products of elliptic curves?” In: Math. Ann. 214 (1975), pp. 35–47. [SGA3] Michael Artin, Jean-Etienne Bertin, Michel Demazure , Alexander Grothendieck, Pierre Gabriel, Michel Raynaud, and Jean-Pierre Serre. Schémas en groupes . Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientiﬁques. ...

work page 1975

[10] [10]

Essential dimension of inﬁnitesi mal commutative unipotent group schemes

[Tos19] Dajano Tossici. “Essential dimension of inﬁnitesi mal commutative unipotent group schemes”. In: Boll. Unione Mat. Ital. 12.4 (2019), pp. 575–581. [TV13] Dajano Tossici and Angelo Vistoli. “On the essential dimension of inﬁnitesimal group schemes”. In: Amer. J. Math. 135.1 (2013), pp. 103–114. Institut de Mathématiques de Bordeaux, 351 Cours de la ...

work page 2019