On the Locus of Smooth Plane Curves with a Fixed Automorphism Group

Badr, Eslam; Bars, Francesc

doi:10.1007/s00009-016-0705-9

Cited by 9 publications

(17 citation statements)

References 10 publications

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“…We consider the following two cases: (Case d = 4). We use quite similar techniques as the ones in [3,4,5]. It is clear that ψ := [X : ζ d/2 Y : Z] ∈ Aut(S t,d ) is an homology of order d/2 ≥ 4 (Definition 4.4).…”

Section: Non-hyperelliptic Real Curves Not Definable Over Their Fieldmentioning

confidence: 99%

Riemann surfaces defined over the reals

2018

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Section: Non-hyperelliptic Real Curves Not Definable Over Their Fieldmentioning

confidence: 99%

Riemann surfaces defined over the reals

2018

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“…Hence, m|d − 1 and we can take a generator (a, b) = (0, 1). Therefore, by checking each monomial's invariance, we obtain that L j,Z = 0 only if j ∈ S(2) m and we get types m, (0, 1) of (1).…”

Section: Cyclic Automorphism Group Of Non-singular Plane Curvesmentioning

confidence: 99%

“…This phenomena is not true in general for an arbitrary m. In other words, with the aid of the algorithm in §2, we prove that ρ(M P l g (Z/m)) has the property of being represented by an unique fixed normal form. But the moduli M P l g (Z/m) with m not "large" or "very large" is not in general given by a single equation with some parameters (counter examples are provided in [1]).…”

Section: Introductionmentioning

confidence: 99%

“…Any element in M P l 6 (Z/8) has, up to K-isomorphism, a plane models of the form X 5 + Y 4 Z + XZ 4 + βX 3 Z 2 = 0 for certain/s β (note that β = ±2 for non-singularity). We constructed in [1] a bijection map…”

Section: Introductionmentioning

confidence: 99%

“…degree 8 + Y 8 + αXZ 7 + β4,4X 4 Y 4 24, (1, 17) X 8 + Y 7 Z + αY Z 7 + β8,4Y 4 Z 416,(1, 9) X 8 + Y 7 Z + αY Z 7 + β8,5Y 5 Z 3 + β8,3Y 3 Z 5 14, (7, 8) X 8 + Y 8 + αXZ 7 + β2,2X 6 Y 2 + β4,4X 4 Y 4 + β6,6X 2 Y 6 12, (1, 5) X 8 + Y 7 Z + αY Z 7 + β8,4Y 4 Z 4 + β4,2X 4 Y 2 Z 2 8, (1, 2) X 8 + Y 8 + Z 8 + β4,0X 4 Z 4 + β5,2X 3 Y 2 Z 3 + β6,4X 2 Y 4 Z 2 + β7,6XY 6 Z 8, (1, 3) X 8 + Y 8 + Z 8 + β4,2X 4 Y 2 Z 2 + β8,4Y 4 Z 4 + X 2 β6,1Y Z 5 + β6,5Y 5 Z 8, (1, 4) X 8 + Y 8 + Z 8 + β2,0X 6 Z 2 + β4,0X 4 Z 4 + β5,4X 3 Y 4 Z + β6,0X 2 Z 6 + β7,4XY 4 Z 3 XZ 7 + αXY 7 + β4,1X 4 Y Z 3 + β5,3X 3 Y 3 Z 2 + β6,5X 2 Y 5 Z + β8,2Y 2 Z 6 7, (1, 3) X 8 + XZ 7 + αXY 7 + β3,1X 5 Y Z 2 + β5,4X 3 Y 4 Z + β6,2X 2 Y 2 Z 4 + β8,5Y 5 Z 3 7, (1, 6) X 8 + XZ 7 + αXY 7 + β2,1X 6 Y Z + β4,2X 4 Y 2 Z 2 + β6,3X 2 Y 3 Z 3 + β8,4Y 4 Z 4 + Y 7 Z + αY Z 7 + β2,1X 6 Y Z + β4,2X 4 Y 2 Z 2 + β8,4Y 4 Z 4 +X 2 β6,0Z 6 + β6,3Y 3 Z 3 + β6,6Y 6 4, (0, 1) Z 8 + Z 4 L4,Z + L8,Z 3, (1, 2) X 8 + Y 7 Z + αY Z 7 + β8,4Y 4 Z 4 + β2,1X 6 Y Z + β4,2X 4 Y 2 Z 2 + X 5 β3,0Z 3 + β3,3Y 3 + +X 3 β5,1Y Z 4 + β5,4Y 4 Z + X 2 β6,0Z 6 + β6,3Y 3 Z 3 + β6,6Y 6 + X β7,2Y 2 Z 5 + β7,5Y 5 Z 2 2, (0, 1) Z 8 + Z 6 L2,Z + Z 4 L4,Z + Z 2 L6,Z + L8,Z…”

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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 twists of smooth plane curves

2018

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Abstract. Given a smooth curve defined over a field k that admits a non-singular plane model over k, a fixed separable closure of k, it does not necessarily have a non-singular plane model defined over the field k. We determine under which conditions this happens and we show an example of such phenomenon: a curve defined over k admitting plane models but none defined over k. Now, even assuming that such a smooth plane model exists, we wonder about the existence of non-singular plane models over k for its twists. We characterize twists possessing such models and we also show an example of a twist not admitting any non-singular plane model over k. As a consequence, we get explicit equations for a non-trivial Brauer-Severi surface. Finally, we obtain a theoretical result to describe all the twists of smooth plane curves with cyclic automorphism group having a model defined over k whose automorphism group is generated by a diagonal matrix.

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On the Locus of Smooth Plane Curves with a Fixed Automorphism Group

Cited by 9 publications

References 10 publications

Riemann surfaces defined over the reals

Riemann surfaces defined over the reals

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

On twists of smooth plane curves

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