Finite group schemes of essential dimension one

Najmuddin Fakhruddin

REVIEW 2 major objections 5 minor 10 references

Finite group schemes of essential dimension one

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A finite group scheme with essential dimension one must embed into $PGL_2$, and all infinitesimal examples are $\alpha_{p^n}$, $\mu_{p^n}$, and (for odd $p$) quadratic twists of $\mu_{p^n}$.

desk verdict A short, careful proof that essential dimension one forces PGL_2 embeddings and yields a clean classification of infinitesimal groups; the one thing to check in refereeing is the char-2 twisted μ_{2^n} dependency. read the letter →

arxiv 1908.02438 v2 pith:JDMSNGSG submitted 2019-08-07 math.AG

Najmuddin Fakhruddin This is my paper · ORCID

classification math.AG MSC 14L2014L3012F05

keywords essentialdimensionfinitegroupschemesinfinitesimalPGL_2actionsoncurvesalpha_{p^n}mu_{p^n}quadratictwists

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a structural constraint: any finite group scheme over a field whose essential dimension is one embeds into the projectivized general linear group $PGL_{2/k}$ and has Lie algebra of dimension at most one. For infinitesimal group schemes, those supported entirely at the identity, this condition is also sufficient, yielding a complete classification: they exist only in positive characteristic and are the groups $\alpha_{p^n}$, the $p$-power roots of unity $\mu_{p^n}$, and, when $p \neq 2$, quadratic twists of $\mu_{p^n}$. Over algebraically closed fields the same criterion classifies all finite group schemes of essential dimension one. The payoff is that a subtle invariant measuring the complexity of group actions is pinned down completely for an entire class of groups.

What carries the argument

The load-bearing mechanism is Proposition 2.2, an extension criterion: if an infinitesimal group scheme $G$ has a generically defined action on a normal projective curve $Y$, and there exists a normal projective variety $X$ with a regular $G$-action together with a dominant $G$-equivariant rational map $X \dashrightarrow Y$, then the action extends uniquely to a regular action on $Y$. This lets the author pass from a versal $G$-torsor over a curve to an honest embedding $G \hookrightarrow \mathrm{Aut}(Y) \cong PGL_{2/k}$. The second ingredient is the structure theory of infinitesimal group schemes with one-dimensional Lie algebra: such a $G$ is either multiplicative or unipotent (Proposition 2.9), and in each case the $PGL_2$ embedding forces a normal form, containment in a torus for multiplicative groups and containment in a Borel subgroup for unipotent groups.

What would settle it

Take $k$ of characteristic $2$ and a non-split quadratic twist $G$ of $\mu_4$ embedded in $PGL_2$ as a subgroup of the non-split torus. Compute $\mathrm{ed}_k(G)$ directly by looking for a one-dimensional compression of its versal torsor: if a $G$-torsor over a rational curve dominates all others, then $\mathrm{ed}_k(G) = 1$ and Theorem 1.1(2) is false; the cited result predicts $\mathrm{ed}_k(G) = 2$, so the computation would settle the matter.

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Extended reading notes

Core claim

The central theorem (Theorem 1.1) has two parts. First, if a finite group scheme $G$ over $k$ has $\mathrm{ed}_k(G) = 1$, then $G$ embeds in $PGL_{2/k}$ and $\dim_k \mathrm{Lie}(G) \leq 1$. Second, for infinitesimal $G$, $\mathrm{ed}_k(G) = 1$ exactly when $G$ embeds in $PGL_{2/k}$, $\dim_k \mathrm{Lie}(G) = 1$, and $G$ lifts to a subgroup scheme of $GL_{2/k}$. The resulting list is $\alpha_{p^n}$ for all $n > 0$, $\mu_{p^n}$ for all $n > 0$, and, for $p \neq 2$, any form of $\mu_{p^n}$ that becomes isomorphic to $\mu_{p^n}$ over a quadratic extension. In characteristic $2$, non-split quadratic twists are excluded by a cited computation showing their $2$-essential dimension is $2$. The proof combines a new extension criterion for rational actions on curves with standard structure theory of finite group schemes.

Load-bearing premise

The classification in characteristic $2$ depends on a cited result, not proved in this paper: a non-trivial quadratic twist of the group of $2^n$-th roots of unity has $2$-essential dimension $2$, and if that computation were wrong some twisted $\mu_{2^n}$ would have essential dimension one and the list in Theorem 1.1(2) would be incomplete.

Editorial extensions

If this is right

If $\mathrm{ed}_k(G) = 1$, the group $G$ acts generically freely on $\mathbb{P}^1_k$, so all essential-dimension-one groups arise from curve quotients $\mathbb{P}^1 \to \mathbb{P}^1/G$.
For infinitesimal groups, essential dimension one can occur only in characteristic $p > 0$; in characteristic $0$ no nontrivial infinitesimal group scheme has essential dimension one.
The $p$-torsion of a supersingular elliptic curve has essential dimension $2$, since it is an extension of $\alpha_p$ by $\alpha_p$ but is not isomorphic to $\alpha_{p^2}$; this answers a question in earlier work on almost-special group schemes.
Any finite commutative unipotent group whose Verschiebung morphism has nilpotence order at least $2$ has essential dimension at least $2$, confirming a conjectured lower bound in that range.
Over a perfect field, a finite group scheme with constant étale quotient has essential dimension at most one exactly when it embeds in $PGL_2$, has Lie algebra dimension at most one, and lifts to $GL_2$ (Theorem 4.1).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same extension criterion is a natural tool for the open question the paper leaves: whether every finite group scheme action on a function field of transcendence degree one extends to a proper model.
If the lift-to-$GL_2$ condition is necessary for every finite group scheme, as the paper shows for constant and infinitesimal groups, then combining Theorem 1.1 with published lists of constant groups would yield a full classification over arbitrary fields; this is the author's stated next step, not carried out here.
The classification suggests a testable pattern: essential dimension one is equivalent to the existence of a faithful action on a rational curve whose quotient has dimension one, a perspective that may transfer to other low-dimensional group actions.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline result is exactly what the title says: a finite group scheme of essential dimension one embeds in PGL_2, and for infinitesimal groups this gives a complete list: α_{p^n}, μ_{p^n}, and — for p≠2 — quadratic twists of μ_{p^n}. The proof is clear and mostly self-contained, with the structure theory of group schemes used in a straightforward way. Proposition 2.2, an extension criterion for rational actions on curves, is the key new tool, and it is nicely motivated by the example showing why the hypothesis is needed.

What the paper does well: it states the limitations honestly, fixes the earlier p=2 error mentioned in the acknowledgements, and gives a useful application (the p-torsion of a supersingular elliptic curve has essential dimension two). The classification of infinitesimal groups is genuinely new; prior work covered constant groups only. The proofs are written out in enough detail that I could follow them without filling in gaps, which is rare for this subject.

Soft spots: the main one is the char-2 multiplicative case. Theorem 1.1(2)(c) excludes non-split quadratic twists of μ_{2^n} by invoking [6, Proposition 6.1], which is cited without being stated or proved. The referee should verify that this proposition really applies to finite flat group schemes of multiplicative type and not just to the ambient torus. If it applies, the classification is safe. If it does not, then part (c) collapses. I have no reason to think the citation is wrong, but because the argument rides entirely on that single external result, it deserves a careful check. The other 2-essential dimension result used in Theorem 4.1 is also cited rather than proved, but that is a less central point.

I did not find circularity. The arguments reduce the classification to standard structure theory and to the cited external results; they do not assume what they are trying to prove. The only genuinely load-bearing external dependency is the char-2 twisted μ_{2^n} bound.

Who this is for: anyone working on essential dimension, finite group schemes, or rational actions on curves. It is a well-written research note, not a survey, and it would serve as a good starting point for further classification work. It deserves a serious referee rather than a desk rejection. I would accept the paper with the expectation that the referee checks the [6] dependency and possibly asks the author to state the needed proposition in the text. No rewrite is needed; the paper is already in publishable shape.

Referee Report

2 major / 5 minor

Summary. The paper studies finite group schemes of essential dimension one over a field k. Theorem 1.1 states that if ed_k(G)=1, then G embeds in PGL_{2/k} and dim_k Lie(G) ≤ 1. For infinitesimal G, it gives an if-and-only-if criterion in terms of an embedding into PGL_2, a one-dimensional Lie algebra, and a lift to GL_2, and it lists the possible groups: α_{p^n}, μ_{p^n}, and, for p ≠ 2, quadratic twists of μ_{p^n} that become split over a quadratic extension. The paper also proves Theorem 4.1, a characterization for perfect fields when the étale quotient is constant, and applies the main theorem to show that the p-torsion of a supersingular elliptic curve has essential dimension two. The main technical novelty is an extension criterion, Proposition 2.2, for rational actions of infinitesimal group schemes on curves.

Significance. If correct, the paper gives a clean and explicit classification of a previously only partially understood class of objects, and it reduces the essential-dimension-one condition to an embedding problem in PGL_2. The proof is concise and mostly self-contained, with the p=2 case explicitly addressed and an earlier error acknowledged. The extension criterion in Proposition 2.2 is a useful tool in its own right. The external result used for the characteristic-2 multiplicative case is, however, a load-bearing dependency that needs to be made fully transparent; with that resolved, the main theorem would be a significant contribution to the essential dimension literature.

major comments (2)

[§3, proof of Theorem 1.1(2)(c), char(k)=2 case] The exclusion of nontrivial quadratic twists of μ_{2^n} in characteristic 2 is delegated to [6, Proposition 6.1], which is neither stated nor reproduced. Since reference [6] is a paper on algebraic tori, it is not evident from the present text that its hypotheses apply to finite flat group schemes of multiplicative type such as a twisted μ_{2^n}. If that proposition concerns only tori, then the assertion ed_k(G;2)=2 for twisted μ_{2^n}, and hence part (2)(c) of Theorem 1.1, has no proof in the paper. Please state the proposition explicitly, verify its applicability to these finite group schemes, or supply a direct proof.
[§3, proof of the Claim in the unipotent case] The sentence 'Since Lie(G) is one dimensional, the action of G on V ... is generically free' is used to invoke versality and obtain a rational equivariant map V → P^1, but no justification is given. The assertion is plausible and likely true, but it is a load-bearing step for the unipotent case; a short argument, for example showing that a nontrivial generic stabilizer would act trivially on k(V) and contradict faithfulness, should be included.

minor comments (5)

[§1, Theorem 1.1(1)] The word 'embeded' should be 'embedded'.
[§2.1, Proposition 2.2] The assertion that the restriction f|_U is surjective is compressed into a parenthetical about complete intersection curves; since this is the only place the curve hypothesis is used, one sentence explaining why f|_C is both proper and dominant would improve clarity.
[§4.1, proof of Theorem 4.1] The phrase 'ed_k(G0)=1 and ed_k(Get)' should read 'ed_k(G0)=1 and ed_k(Get)=1'.
[§1.1] The paper uses p-essential dimension ed_k(G;p) in §3 but does not define it; a one-line definition or an explicit pointer to [7, §3d] would make the note more self-contained.
[§2.1, Proposition 2.2] There are typographical artifacts such as '/axisshort/axisshort/arrowaxisright' in the displayed statement; these should be replaced by proper arrow notation in the published version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof derives the classification from structural group-scheme theorems and independent external results, with no fitted parameters or self-citation load-bearing steps.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.1 is proved using Proposition 2.2 (an extension criterion proved in the paper via Lemma 2.1), structural facts about group schemes (cited to SGA 3 and Demazure-Gabriel), and the theory of essential dimension. The classification of multiplicative subgroups in characteristic 2 does rely on the external bound ed_k(G;2)=2 for non-trivial quadratic twists of μ_{2^n}, quoted as [6, Proposition 6.1] from a paper by different authors. That is a genuine verification dependency, but it is not circular: the cited result is independent prior work, not a self-citation, and it is not equivalent to the paper's conclusion. No step fits any of the enumerated circularity patterns: there is no self-definitional construction, no fitted input renamed as a prediction, no load-bearing self-citation, no uniqueness imported from the authors' own prior work, and no ansatz smuggled in via citation. The note itself flags the limitation that [6, Proposition 6.1] is not proved here, but that is a correctness/verification caveat, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a pure mathematics proof. It introduces no free parameters or invented entities. It relies on standard theorems in the theory of group schemes and on one cited essential-dimension result for quadratic twists. These are listed above; none of them is circular with the main claim.

assumptions (5)

domain assumption Classification and structure theory of finite group schemes over a field (Demazure-Gabriel [4]), including semidirect product decomposition over perfect fields and the theory of height one subgroups.
Used throughout Sections 2 and 3 to reduce to multiplicative/unipotent groups and to construct lifts.
domain assumption SGA 3 results: the centralizer of a group scheme of multiplicative type in a smooth affine group scheme is smooth ([1, XI, Cor 2.4]); extensions of μ_p by α_p over an algebraically closed field are trivial ([1, Thm 6.1.1(B)]).
Used in Proposition 2.9 to show a height-one-Lie-algebra infinitesimal group is either multiplicative or unipotent and in the proof of Theorem 1.1 to control centralizers.
domain assumption [6, Proposition 6.1]: a non-trivial quadratic twist of μ_{2^n} has 2-essential dimension 2 over a field of characteristic 2.
This cited theorem is the load-bearing external result in the char 2 multiplicative classification; it excludes non-split forms of μ_{2^n} from ed=1.
domain assumption Existence of versal torsors and the definition/compression properties of essential dimension (Merkurjev [7], Reichstein [8]).
The proof of the main theorem starts from a versal G-torsor over a rational variety and compresses it to a torsor over a curve.
standard math Lüroth's theorem: a field of transcendence degree one over k that is a subfield of k(t) is itself k(t).
Used in the proof of Theorem 1.1 to conclude the normal projective curve Y with function field k(T) is P^1_k.

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Cite this review

Pith. "Pith review of Finite group schemes of essential dimension one." pith.science (2026). https://pith.science/paper/JDMSNGSG

@misc{pith2026190802438,
  author       = {Pith},
  title        = {Pith review of: Finite group schemes of essential dimension one},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JDMSNGSG}},
  note         = {Machine review of arXiv:1908.02438}
}

abstract

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any field and a characterisation of all finite group schemes of essential dimension one over algebraically closed fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[6]

L ¨otscher, M

R. L ¨otscher, M. MacDonald, A. Meyer, and Z. Reichstein , Essential dimension of algebraic tori, J. Reine Angew. Math., 677 (2013), pp. 1–13. 5

work page 2013
[1]

II: Groupes de type multiplicatif, et s tructure des sch´ emas en groupes g´ en´ eraux, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962/64 (SGA 3)

Sch´ emas en groupes. II: Groupes de type multiplicatif, et s tructure des sch´ emas en groupes g´ en´ eraux, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962/64 (SGA 3). Dirig´ e par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer- Verlag, Berlin, 1962/1964. 4, 5

work page 1962
[2]

Babic and V

A. Babic and V. Chernousov , Lower bounds for essential dimensions in characteristic 2 v ia orthog- onal representations, Paciﬁc J. Math., 279 (2015), pp. 37–63. 7

work page 2015
[3]

Chu, S.-J

H. Chu, S.-J. Hu, M.-C. Kang, and J. Zhang , Groups with essential dimension one , Asian J. Math., 12 (2008), pp. 177–191. 1, 6

work page 2008
[4]

Demazure and P

M. Demazure and P. Gabriel , Groupes alg´ ebriques. Tome I: G´ eom´ etrie alg´ ebrique, g´en´ eralit´ es, groupes commutatifs, Masson & Cie, ´Editeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice Corps de classes local par Michiel Hazewinkel. 2, 3, 4, 6

work page 1970
[5]

Ledet , Finite groups of essential dimension one , J

A. Ledet , Finite groups of essential dimension one , J. Algebra, 311 (2007), pp. 31–37. 1, 7 2This, together with the reference in [ 2], was pointed out by Z. Reichstein. 8 NAJMUDDIN F AKHRUDDIN

work page 2007
[7]

A. S. Merkurjev , Essential dimension: a survey , Transform. Groups, 18 (2013), pp. 415–481. 1, 2

work page 2013
[8]

Reichstein , Essential dimension , in Proceedings of the International Congress of Mathematicians

Z. Reichstein , Essential dimension , in Proceedings of the International Congress of Mathematicians . Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188. 1, 2

work page 2010

Show all 10 references

[9]

Tossici , Essential dimension of iniﬁnitesimal commutative unipote nt group schemes , Bollettino dell’Unione Matematica Italiana, (2019)

D. Tossici , Essential dimension of iniﬁnitesimal commutative unipote nt group schemes , Bollettino dell’Unione Matematica Italiana, (2019). 1

2019
[10]

Tossici and A

D. Tossici and A. Vistoli , On the essential dimension of inﬁnitesimal group schemes , Amer. J. Math., 135 (2013), pp. 103–114. 2, 7 School of Mathematics, Tata Institute of Fundamental Resea rch, Homi Bhabha Road, Mumbai 400005, India E-mail address : naf@math.tifr.res.in

2013

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

L ¨otscher, M

R. L ¨otscher, M. MacDonald, A. Meyer, and Z. Reichstein , Essential dimension of algebraic tori, J. Reine Angew. Math., 677 (2013), pp. 1–13. 5

work page 2013

[2] [1]

II: Groupes de type multiplicatif, et s tructure des sch´ emas en groupes g´ en´ eraux, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962/64 (SGA 3)

Sch´ emas en groupes. II: Groupes de type multiplicatif, et s tructure des sch´ emas en groupes g´ en´ eraux, S´ eminaire de G´ eom´ etrie Alg´ ebrique du Bois Marie 1962/64 (SGA 3). Dirig´ e par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer- Verlag, Berlin, 1962/1964. 4, 5

work page 1962

[3] [2]

Babic and V

A. Babic and V. Chernousov , Lower bounds for essential dimensions in characteristic 2 v ia orthog- onal representations, Paciﬁc J. Math., 279 (2015), pp. 37–63. 7

work page 2015

[4] [3]

Chu, S.-J

H. Chu, S.-J. Hu, M.-C. Kang, and J. Zhang , Groups with essential dimension one , Asian J. Math., 12 (2008), pp. 177–191. 1, 6

work page 2008

[5] [4]

Demazure and P

M. Demazure and P. Gabriel , Groupes alg´ ebriques. Tome I: G´ eom´ etrie alg´ ebrique, g´en´ eralit´ es, groupes commutatifs, Masson & Cie, ´Editeur, Paris; North-Holland Publishing Co., Amsterdam, 1970. Avec un appendice Corps de classes local par Michiel Hazewinkel. 2, 3, 4, 6

work page 1970

[6] [5]

Ledet , Finite groups of essential dimension one , J

A. Ledet , Finite groups of essential dimension one , J. Algebra, 311 (2007), pp. 31–37. 1, 7 2This, together with the reference in [ 2], was pointed out by Z. Reichstein. 8 NAJMUDDIN F AKHRUDDIN

work page 2007

[7] [7]

A. S. Merkurjev , Essential dimension: a survey , Transform. Groups, 18 (2013), pp. 415–481. 1, 2

work page 2013

[8] [8]

Reichstein , Essential dimension , in Proceedings of the International Congress of Mathematicians

Z. Reichstein , Essential dimension , in Proceedings of the International Congress of Mathematicians . Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 162–188. 1, 2

work page 2010