On the automorphism group of non-singular plane curves fixing the degree

Badr, Eslam; Bars, Francesc

doi:10.48550/arxiv.1503.01149

Cited by 2 publications

(12 citation statements)

References 9 publications

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“…For d even, we prove in section § 5 that M P l 10 (Z/3Z) is not ES-irreducible. It is to be noted that we may conjecture, by our work in [1], that the locus M P l g (Z/mZ) could not be ES-Irreducible only if m divides d or d − 1 (this is true at least until degree 9 by [1]). Concerning positive characteristic, in the last section ( § 6) of this paper, we prove that the above examples of non-irreducible loci are also valid when K is an algebraically closed field of positive characteristic p > 0, provided that the characteristic p is big enough, once we fix the degree d.…”

Section: Introductionmentioning

confidence: 88%

“…where 0 ≤ a < b, and ξ m a primitive m-th root of unity in K. Following the same technique in [8] or in [1] (for a general discussion), we can associate to the set ρ(M P l g (< σ >) a non-singular plane equation F m,(a,b) (X; Y ; Z) with a certain set of parameters (under some algebraic restrictions in order to ensure the non-singularity). This is a "normal form" for ρ(M P l g (< σ >)), and it is also unique(up to K-equivalence) by construction.…”

Section: Fix a [ρ]mentioning

confidence: 99%

“…We also investigated such restrictions particularly for the loci ρ(M P l 6 (Z/8)) and ρ( M P l 6 (Z/8)) at the end of this section. It is difficult to determine the groups G and [ρ] ∈ A G such that ρ(M P l g (G)) is non-empty for some fixed g. Henn [12] obtained this determination for g = 3, Badr-Bars [2] for g = 6, and for a general implementation of any degree d ≥ 5, we refer to [1] (in which we formulate an algorithm to determine the ρ's when G is cyclic). Definition 2.5.…”

Section: Fix a [ρ]mentioning

confidence: 99%

“…As an explicit example, we deal with the question for the locus M P l 6 (Z/8), which is ES-Irreducible, and is represented by a single normal form with only one paramater. In [1], we proved that M P l g (G) is irreducible when G has an element of order (d − 1) 2 , d(d − 1), d(d − 2) or d 2 − 3d + 3, since this locus has only one element. In particular, we prove in [1] that M P l g (Z/d(d − 1)) and M P l g (Z/(d − 1) 2 ) are irreducible.…”

Section: Introductionmentioning

confidence: 99%

“…In [1], we proved that M P l g (G) is irreducible when G has an element of order (d − 1) 2 , d(d − 1), d(d − 2) or d 2 − 3d + 3, since this locus has only one element. In particular, we prove in [1] that M P l g (Z/d(d − 1)) and M P l g (Z/(d − 1) 2 ) are irreducible.…”

Section: Introductionmentioning

confidence: 99%

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On the locus of smooth plane curves with a fixed automorphism group

Badr,

Bars

2015

Preprint

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Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is isomorphic to a subgroup of Aut(δ), the full automorphism group of δ, and let Mg(G) be the subset of curves δ such that G ∼ = Aut(δ). Now, for an integer d ≥ 4, let M P l g be the subset of Mg representing smooth, genus g plane curves of degree d (in such case, g = (d − 1)(d − 2)/2), and consider the sets M P l g (G) := M P l g ∩ Mg(G) and M P l g (G) := Mg(G) ∩ M P l g . In this paper, we study some aspects of the irreducibility of M P l g (G) and its interrelation with the existence of "normal forms", i.e. non-singular plane equations (depending on a set of parameters) such that a specialization of the parameters gives a certain non-singular plane model associated to the elements of M P l g (G). In particular, we introduce the concept of being equation strongly irreducible (ES-Irreducible) for which the locus M P l g (G) is represented by a single "normal form". Henn, in [12], and in [14], observed that M P l 3 (G) is ES-Irreducible. In this paper we prove that this phenomena does not occur for any odd d > 4. More precisely, let Z/mZ be the cyclic group of order m, we prove that M P l g (Z/(d − 1)Z) is not ES-Irreducible for any odd integer d ≥ 5, and the number of its irreducible components is at least two. Furthermore, we conclude the previous result when d = 6 for the locus M P l 10 (Z/3Z). Lastly, we prove the analogy of these statements when K is any algebraically closed field of positive characteristic p such that p > (d − 1)(d − 2) + 1.

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Section: Introductionmentioning

confidence: 88%

Section: Fix a [ρ]mentioning

confidence: 99%

Section: Fix a [ρ]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the locus of smooth plane curves with a fixed automorphism group

Badr,

Bars

2015

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Non-singular plane curves with an element of “large” order in its automorphism group

Badr

Bars

2016

Int. J. Algebra Comput.

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Let Mg be the moduli space of smooth, genus g curves over an algebraically closed field K of zero characteristic. Denote by Mg(G) the subset of Mg of curves δ such that G (as a finite non-trivial group) is 2010 Mathematics Subject Classification. 14H37, 14H50, 14H45. ) >.(3) if σ has order d(d − 2) then δ is K-isomorphic to X d + Y d−1 Z + Y Z d−1 = 0 and for d = 4, 6 we haveis an irreducible locus with one element, andThe automorphism groups for d = 4, 6 are given explicitly in §3.3, Proposition 15.(4) if σ has order d 2 − 3d + 3 then δ is K-isomorphic to the Klein curve K d :) is irreducible with one element, andWe refer to Remark 18 of §3.4 for the classical case d = 4.Remark 2. The above situations do not fit with curves that have large automorphism group in the classical definition. For example, the curve δ : X d + Y d−1 Z + XZ d−1 = 0 is defined over Q, δ/Aut(δ) is a projective line and the morphism δ → δ/Aut(δ) is ramified at two points of ramification index (d − 1) 2 and at d − 1-points of

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On the automorphism group of non-singular plane curves fixing the degree

Cited by 2 publications

References 9 publications

On the locus of smooth plane curves with a fixed automorphism group

On the locus of smooth plane curves with a fixed automorphism group

Non-singular plane curves with an element of “large” order in its automorphism group

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