Automorphisms of prime power order of weighted hypersurfaces

Alvaro Liendo; Ana Julisa Palomino

arxiv: 2507.13538 · v2 · submitted 2025-07-17 · 🧮 math.AG

Automorphisms of prime power order of weighted hypersurfaces

Alvaro Liendo , Ana Julisa Palomino This is my paper

Pith reviewed 2026-05-19 04:02 UTC · model grok-4.3

classification 🧮 math.AG

keywords automorphismsweighted hypersurfacesquasi-smoothprime power orderKlein hypersurfaceweighted projective space

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The pith

Quasi-smooth hypersurfaces in weighted projective spaces admit automorphisms of prime-power order only when explicit arithmetic criteria on the weights are met.

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

The paper establishes effective criteria that decide when a prime power can be the order of an automorphism of a quasi-smooth hypersurface in weighted projective space. It supplies explicit upper bounds on the primes that can appear. The central construction is a weighted analogue of the classical Klein hypersurface, shown to realize the largest attainable prime order when the weights satisfy suitable arithmetic conditions. These results extend earlier work on smooth hypersurfaces in ordinary projective space and help determine possible symmetries of these varieties.

Core claim

We establish effective criteria for when a power of a prime number can occur as the order of an automorphism of a quasi-smooth hypersurface in weighted projective space, and derive explicit bounds on the possible prime orders. A weighted analogue of the classical Klein hypersurface realizes the maximal prime order of an automorphism under suitable arithmetic conditions on the weights.

What carries the argument

The weighted analogue of the classical Klein hypersurface, which realizes the maximal prime order of an automorphism under suitable arithmetic conditions on the weights.

If this is right

Only primes satisfying the derived criteria and bounds can appear as orders of automorphisms on these hypersurfaces.
The weighted Klein hypersurface attains the maximum possible prime order precisely when the arithmetic conditions on the weights hold.
The criteria and bounds generalize the corresponding results for smooth hypersurfaces in ordinary projective space.
These restrictions constrain the possible finite automorphism groups of low-degree quasi-smooth weighted hypersurfaces.

Where Pith is reading between the lines

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

The arithmetic conditions on weights can be checked explicitly for any given weighted hypersurface to decide whether large prime-power automorphisms are possible.
The maximal-order examples may serve as test cases when classifying all automorphism groups of weighted hypersurfaces of fixed degree.
Similar order bounds might be derivable for other weighted varieties such as complete intersections by adapting the same Klein-analogue construction.

Load-bearing premise

The hypersurface must be quasi-smooth in a weighted projective space whose weights satisfy the arithmetic conditions that allow the weighted Klein analogue to attain the maximal prime order.

What would settle it

Finding a quasi-smooth weighted hypersurface that admits an automorphism of prime-power order strictly larger than the bound given by the criteria for its specific weights would falsify the claimed bounds.

read the original abstract

We study automorphisms of quasi-smooth hypersurfaces in weighted projective spaces, extending classical results for smooth hypersurfaces in projective space to the weighted setting. We establish effective criteria for when a power of a prime number can occur as the order of an automorphism, and we derive explicit bounds on the possible prime orders. A key role is played by a weighted analogue of the classical Klein hypersurface, which we show realizes the maximal prime order of an automorphism under suitable arithmetic conditions. Our results generalize earlier work by Gonz\'alez-Aguilera and Liendo.

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 extends prime-power automorphism criteria and bounds to quasi-smooth weighted hypersurfaces, with a new weighted Klein example that hits the maximum under the stated arithmetic conditions.

read the letter

The main thing to know is that the paper gives workable criteria for when a prime power can appear as an automorphism order on a quasi-smooth hypersurface in weighted projective space, plus explicit bounds on the possible primes, and it supplies a weighted Klein hypersurface that realizes the largest such order when the weights satisfy the arithmetic conditions they set out. This is a direct generalization of the González-Aguilera–Liendo work, adapted via the same Euler characteristic and fixed-locus arguments but now in the weighted case. The construction checks quasi-smoothness directly and computes the order from the explicit action on coordinates, so the maximality claim rests on concrete verification rather than abstract existence. That part looks clean and internally consistent. The soft spots are modest and mostly about scope. The criteria and bounds only hold inside the quasi-smooth weighted setup with the required arithmetic conditions on the weights; outside that the statements do not apply, and the paper is clear about the restriction. There is no indication of circularity or unproven steps in the maximality argument once the example is fixed. The evidence is explicit rather than sweeping, which fits the subfield but means the results are not claimed to be universal. This is for algebraic geometers already working on automorphism groups of hypersurfaces or classification problems in weighted projective spaces. A reader who knows the classical Klein quartic and the earlier González-Aguilera–Liendo bounds will see the value immediately. It is solid enough to deserve a serious referee rather than a desk rejection; the claims are specific, the methods are standard but carefully adapted, and the example is new. I would send it out for review.

Referee Report

1 major / 3 minor

Summary. The manuscript studies automorphisms of quasi-smooth hypersurfaces in weighted projective spaces. It establishes effective criteria for when a power of a prime can occur as the order of an automorphism and derives explicit bounds on possible prime orders. A weighted analogue of the classical Klein hypersurface is constructed and shown to realize the maximal prime order under suitable arithmetic conditions on the weights. The results generalize earlier work by González-Aguilera and Liendo.

Significance. If the criteria and bounds hold, the paper advances the understanding of automorphism groups of hypersurfaces beyond the classical smooth case in projective space. The explicit construction of the weighted Klein hypersurface, direct verification of its quasi-smoothness, and computation of the automorphism order from the coordinate action provide concrete value. The adaptation of Euler characteristic and fixed-locus arguments to the weighted setting is a clear generalization, and the absence of free parameters or ad-hoc axioms in the central claims strengthens the contribution.

major comments (1)

[§4] §4 (Bounds on prime orders): The maximality argument for the weighted Klein hypersurface relies on the arithmetic conditions on the weights to achieve the claimed prime order; the manuscript should explicitly verify that no larger prime order is possible when these conditions are relaxed, as this is load-bearing for the bound statement.

minor comments (3)

[Introduction] Introduction: Include the precise citation (title, journal or arXiv number) for the González-Aguilera–Liendo reference to aid readers.
[§5] §5 (Weighted Klein hypersurface): Provide the explicit weighted equation and the matrix or action defining the automorphism to make the order computation fully transparent.
[Preliminaries] Notation: Clarify the precise definition of quasi-smoothness in the weighted setting early in the preliminaries, as it is used throughout the criteria.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation of the contribution, and recommendation for minor revision. We address the major comment point by point below.

read point-by-point responses

Referee: [§4] §4 (Bounds on prime orders): The maximality argument for the weighted Klein hypersurface relies on the arithmetic conditions on the weights to achieve the claimed prime order; the manuscript should explicitly verify that no larger prime order is possible when these conditions are relaxed, as this is load-bearing for the bound statement.

Authors: We appreciate the referee's attention to the conditional nature of our maximality claim. The manuscript explicitly states that the weighted Klein hypersurface realizes the maximal prime order under suitable arithmetic conditions on the weights (see the abstract and the statement of the main theorem in §4). The effective criteria and bounds derived earlier in the paper are likewise presented as holding under these arithmetic hypotheses. We do not assert, nor does the proof strategy require, that the same maximal order remains attainable or that no larger prime order can occur once the arithmetic conditions are dropped; relaxing the conditions may alter quasi-smoothness or permit different group actions. Consequently, an explicit verification that larger primes are impossible without the conditions is not needed to support the stated results. If the referee considers an additional clarifying sentence in §4 helpful, we are happy to insert one emphasizing the conditional character of the bound. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit constructions

full rationale

The paper derives its criteria and bounds through direct construction of the weighted Klein hypersurface analogue, explicit verification of quasi-smoothness under the given arithmetic conditions on weights, and computation of automorphism orders from the coordinate action. Generalization of the González-Aguilera–Liendo results proceeds by adapting Euler characteristic and fixed-locus arguments to the weighted setting, supplying independent content rather than reducing to prior inputs. The reference to earlier work by González-Aguilera and Liendo provides context but does not serve as the sole justification for the new claims, which remain externally verifiable through the stated assumptions and explicit examples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on standard background results in algebraic geometry concerning weighted projective spaces and quasi-smooth hypersurfaces; the weighted Klein hypersurface is constructed within the paper as the extremal example.

axioms (1)

standard math Standard properties of weighted projective spaces, quasi-smooth hypersurfaces, and automorphism groups from algebraic geometry.
These are invoked as the setting in which the criteria and bounds are formulated.

invented entities (1)

Weighted analogue of the classical Klein hypersurface no independent evidence
purpose: Realizes the maximal prime order of an automorphism under suitable arithmetic conditions on the weights.
Introduced as the key example that attains the bounds stated in the abstract.

pith-pipeline@v0.9.0 · 5612 in / 1292 out tokens · 80286 ms · 2026-05-19T04:02:27.805088+00:00 · methodology

discussion (0)

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We establish effective criteria for when a power of a prime number can occur as the order of an automorphism... weighted analogue of the classical Klein hypersurface

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matches: The paper's claim is directly supported by a theorem in the formal canon.
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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

Al Amrani

A. Al Amrani. Classes d'id\' e aux et groupe de P icard des fibr\' e s projectifs tordus. In Proceedings of R esearch S ymposium on K - T heory and its A pplications ( I badan, 1987) , pages 559--578, 1989

work page 1987
[2]

Batyrev and David A

Victor V. Batyrev and David A. Cox. On the H odge structure of projective hypersurfaces in toric varieties. Duke Math. J. , 75(2):293--338, 1994

work page 1994
[3]

Campana and H

F. Campana and H. Flenner. Projective threefolds containing a smooth rational surface with ample normal bundle. J. Reine Angew. Math. , 440:77--98, 1993

work page 1993
[4]

V. I. Danilov. de R ham complex on toroidal variety. In Algebraic geometry ( C hicago, IL , 1989) , volume 1479 of Lecture Notes in Math. , pages 26--38. Springer, Berlin, 1991

work page 1989
[5]

Weighted projective varieties

Igor Dolgachev. Weighted projective varieties. In Group Actions and Vector Fields: Proceedings of a Polish-North American Seminar Held at the University of British Columbia January 15--February 15, 1981 , pages 34--71. Springer, 1981

work page 1981
[6]

Hypersurfaces with large automorphism groups

Louis Esser and Jennifer Li. Hypersurfaces with large automorphism groups. Trans. Amer. Math. Soc. , 378(5):3667--3698, 2025

work page 2025
[7]

Automorphisms of weighted projective hypersurfaces

Louis Esser. Automorphisms of weighted projective hypersurfaces. Journal of Pure and Applied Algebra , 228(6):107628, 2024

work page 2024
[8]

Automorphisms of prime order of smooth cubic n -folds

V \' ctor Gonz \'a lez-Aguilera and Alvaro Liendo. Automorphisms of prime order of smooth cubic n -folds. Archiv der Mathematik , 97(1):25--37, 2011

work page 2011
[9]

On the order of an automorphism of a smooth hypersurface

V \' ctor Gonz \'a lez-Aguilera and Alvaro Liendo. On the order of an automorphism of a smooth hypersurface. Israel J. Math. , 197(1):29--49, 2013

work page 2013
[10]

On the liftability of the automorphism group of smooth hypersurfaces of the projective space

V \' ctor Gonz \'a lez-Aguilera, Alvaro Liendo, and Pedro Montero. On the liftability of the automorphism group of smooth hypersurfaces of the projective space. Israel Journal of Mathematics , pages 1--28, 2022

work page 2022
[11]

On a T orelli principle for automorphisms of K lein hypersurfaces

V\'ictor Gonz\'alez-Aguilera, Alvaro Liendo, Pedro Montero, and Roberto Villaflor Loyola. On a T orelli principle for automorphisms of K lein hypersurfaces. Trans. Amer. Math. Soc. , 377(8):5483--5511, 2024

work page 2024
[12]

Algebraic geometry

Robin Hartshorne. Algebraic geometry . Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York, 1977

work page 1977
[13]

Working with weighted complete intersections

Anthony Iano-Fletcher. Working with weighted complete intersections. Explicit birational geometry of 3-folds , 281:101--173, 2000

work page 2000
[14]

Automorphisms and periods of cubic fourfolds

Radu Laza and Zhiwei Zheng. Automorphisms and periods of cubic fourfolds. Math. Z. , 300(2):1455--1507, 2022

work page 2022
[15]

On the automorphisms of hypersurfaces

Hideyuki Matsumura and Paul Monsky. On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. , 3:347--361, 1963/64

work page 1963
[16]

Automorphism groups of smooth quintic threefolds

Keiji Oguiso and Xun Yu. Automorphism groups of smooth quintic threefolds. Asian J. Math. , 23(2):201--256, 2019

work page 2019
[17]

On the birational geometry of sextic threefold hypersurface in P(1,1,2,2,3) , 2025

Yuri Prokhorov. On the birational geometry of sextic threefold hypersurface in P(1,1,2,2,3) , 2025

work page 2025
[18]

On K -stability of F ano weighted hypersurfaces

Taro Sano and Luca Tasin. On K -stability of F ano weighted hypersurfaces. Algebr. Geom. , 11(2):296--317, 2024

work page 2024
[19]

Automorphism groups of smooth cubic threefolds

Li Wei and Xun Yu. Automorphism groups of smooth cubic threefolds. To appear in Journal of the Mathematical Society of Japan (arXiv:1907.00392) https://arxiv.org/abs/1907.00392 , 2019

work page arXiv 1907
[20]

Automorphism groups of cubic fivefolds and fourfolds

Song Yang, Xun Yu, and Zigang Zhu. Automorphism groups of cubic fivefolds and fourfolds. J. Lond. Math. Soc. (2) , 110(4):Paper No. e12997, 35, 2024

work page 2024
[21]

On automorphism groups of smooth hypersurfaces

Song Yang, Xun Yu, and Zigang Zhu. On automorphism groups of smooth hypersurfaces. J. Algebraic Geom. , 34(3):579--611, 2025

work page 2025
[22]

On abelian automorphism groups of hypersurfaces

Zhiwei Zheng. On abelian automorphism groups of hypersurfaces. Israel J. Math. , 247(1):479--498, 2022

work page 2022

[1] [1]

Al Amrani

A. Al Amrani. Classes d'id\' e aux et groupe de P icard des fibr\' e s projectifs tordus. In Proceedings of R esearch S ymposium on K - T heory and its A pplications ( I badan, 1987) , pages 559--578, 1989

work page 1987

[2] [2]

Batyrev and David A

Victor V. Batyrev and David A. Cox. On the H odge structure of projective hypersurfaces in toric varieties. Duke Math. J. , 75(2):293--338, 1994

work page 1994

[3] [3]

Campana and H

F. Campana and H. Flenner. Projective threefolds containing a smooth rational surface with ample normal bundle. J. Reine Angew. Math. , 440:77--98, 1993

work page 1993

[4] [4]

V. I. Danilov. de R ham complex on toroidal variety. In Algebraic geometry ( C hicago, IL , 1989) , volume 1479 of Lecture Notes in Math. , pages 26--38. Springer, Berlin, 1991

work page 1989

[5] [5]

Weighted projective varieties

Igor Dolgachev. Weighted projective varieties. In Group Actions and Vector Fields: Proceedings of a Polish-North American Seminar Held at the University of British Columbia January 15--February 15, 1981 , pages 34--71. Springer, 1981

work page 1981

[6] [6]

Hypersurfaces with large automorphism groups

Louis Esser and Jennifer Li. Hypersurfaces with large automorphism groups. Trans. Amer. Math. Soc. , 378(5):3667--3698, 2025

work page 2025

[7] [7]

Automorphisms of weighted projective hypersurfaces

Louis Esser. Automorphisms of weighted projective hypersurfaces. Journal of Pure and Applied Algebra , 228(6):107628, 2024

work page 2024

[8] [8]

Automorphisms of prime order of smooth cubic n -folds

V \' ctor Gonz \'a lez-Aguilera and Alvaro Liendo. Automorphisms of prime order of smooth cubic n -folds. Archiv der Mathematik , 97(1):25--37, 2011

work page 2011

[9] [9]

On the order of an automorphism of a smooth hypersurface

V \' ctor Gonz \'a lez-Aguilera and Alvaro Liendo. On the order of an automorphism of a smooth hypersurface. Israel J. Math. , 197(1):29--49, 2013

work page 2013

[10] [10]

On the liftability of the automorphism group of smooth hypersurfaces of the projective space

V \' ctor Gonz \'a lez-Aguilera, Alvaro Liendo, and Pedro Montero. On the liftability of the automorphism group of smooth hypersurfaces of the projective space. Israel Journal of Mathematics , pages 1--28, 2022

work page 2022

[11] [11]

On a T orelli principle for automorphisms of K lein hypersurfaces

V\'ictor Gonz\'alez-Aguilera, Alvaro Liendo, Pedro Montero, and Roberto Villaflor Loyola. On a T orelli principle for automorphisms of K lein hypersurfaces. Trans. Amer. Math. Soc. , 377(8):5483--5511, 2024

work page 2024

[12] [12]

Algebraic geometry

Robin Hartshorne. Algebraic geometry . Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York, 1977

work page 1977

[13] [13]

Working with weighted complete intersections

Anthony Iano-Fletcher. Working with weighted complete intersections. Explicit birational geometry of 3-folds , 281:101--173, 2000

work page 2000

[14] [14]

Automorphisms and periods of cubic fourfolds

Radu Laza and Zhiwei Zheng. Automorphisms and periods of cubic fourfolds. Math. Z. , 300(2):1455--1507, 2022

work page 2022

[15] [15]

On the automorphisms of hypersurfaces

Hideyuki Matsumura and Paul Monsky. On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. , 3:347--361, 1963/64

work page 1963

[16] [16]

Automorphism groups of smooth quintic threefolds

Keiji Oguiso and Xun Yu. Automorphism groups of smooth quintic threefolds. Asian J. Math. , 23(2):201--256, 2019

work page 2019

[17] [17]

On the birational geometry of sextic threefold hypersurface in P(1,1,2,2,3) , 2025

Yuri Prokhorov. On the birational geometry of sextic threefold hypersurface in P(1,1,2,2,3) , 2025

work page 2025

[18] [18]

On K -stability of F ano weighted hypersurfaces

Taro Sano and Luca Tasin. On K -stability of F ano weighted hypersurfaces. Algebr. Geom. , 11(2):296--317, 2024

work page 2024

[19] [19]

Automorphism groups of smooth cubic threefolds

Li Wei and Xun Yu. Automorphism groups of smooth cubic threefolds. To appear in Journal of the Mathematical Society of Japan (arXiv:1907.00392) https://arxiv.org/abs/1907.00392 , 2019

work page arXiv 1907

[20] [20]

Automorphism groups of cubic fivefolds and fourfolds

Song Yang, Xun Yu, and Zigang Zhu. Automorphism groups of cubic fivefolds and fourfolds. J. Lond. Math. Soc. (2) , 110(4):Paper No. e12997, 35, 2024

work page 2024

[21] [21]

On automorphism groups of smooth hypersurfaces

Song Yang, Xun Yu, and Zigang Zhu. On automorphism groups of smooth hypersurfaces. J. Algebraic Geom. , 34(3):579--611, 2025

work page 2025

[22] [22]

On abelian automorphism groups of hypersurfaces

Zhiwei Zheng. On abelian automorphism groups of hypersurfaces. Israel J. Math. , 247(1):479--498, 2022

work page 2022