Automorphism Groups of Nonsingular Plane Curves of Degree 5

Badr, Eslam; Bars, Francesc

doi:10.1080/00927872.2015.1087547

Cited by 21 publications

(43 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then Hess * does not appear as the full automorphism group of C, except possibly for d = 7 and * = 216. But, following the same discussion that we did for [3,Proposition 12], one easily deduces that there exists no smooth k-plane curve of degree 7, whose automorphism group is conjugate to Hess 216 .…”

Section: 2supporting

confidence: 61%

See 1 more Smart Citation

A note on the stratification by automorphisms of smooth plane curves of genus 6

Badr¹,

García²

2020

Colloq. Math.

Self Cite

View full text Add to dashboard Cite

In this note, we give a so-called representative classification for the strata by automorphism group of smooth k-plane curves of genus 6, where k is a fixed separable closure of a field k of characteristic p = 0 or p > 13. We start with a classification already obtained in [3] and we mimic the techniques in [10] and [11].Interestingly, in the way to get these families for the different strata, we find two remarkable phenomenons that did not appear before. One is the existence of a non 0-dimensional final stratum of plane curves. At a first sight it may sound odd, but we will see that this is a normal situation for higher degrees and we will give a explanation for it.We explicitly describe representative families for all strata, except for the stratum with automorphism group Z/5Z. Here we find the second difference with the lower genus cases where the previous techniques do not fully work. Fortunately, we are still able to prove the existence of such family by applying a version of Lüroth's theorem in dimension 2.

show abstract

Section: 2supporting

confidence: 61%

“…We use the notation M P l g (G) for the subset of M P l g (G) whose elements satisfies G ∼ = Aut(C). The finite groups G for which M P l 6 (G) is non-empty are determined in [3]. Moreover, geometrically complete families that depend on a fixed injective representation ̺ : Aut(C) ֒→ PGL 3 (k) are given.…”

Section: 2mentioning

confidence: 99%

A note on the stratification by automorphisms of smooth plane curves of genus 6

Badr¹,

García²

2020

Colloq. Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…By Theorem 2.1, the map g : C k → D ∼ = P 2 k defined over k defines a non-singular plane model of C over k. Corollary 2.3. Consider a field k such that Br(k) [3] is trivial, where Br(k) [3] denotes the 3-torsion of Br(k). Then any smooth plane curve C over k, admits a non-singular plane model over k, and in particular any twist of C over k admits also a non-singular plane model over k.…”

Section: The Field Of Definition Of a Non-singular Plane Modelmentioning

confidence: 99%

“…It is well-known that the complete linear series g 2 d is unique up to conjugation in PGL 3 (k), the automorphism group of P 2 k , see [11,Lemma 11.28]. Therefore, any k-model of C is defined by F P −1 C (X, Y, Z) := F (P (X, Y, Z)) = 0 for some P ∈ PGL 3 (k), observe that the k-model is an equation in P 2 corresponding to the curve P −1 C which is k-isomorphic to C. We say that C is a smooth plane curve over k if it is k-isomorphic to F Q −1 C (X, Y, Z) = 0 for some Q ∈ PGL 3…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, for smooth plane curves defined over k with a cyclic automorphism group generated by a diagonal matrix, we provide a general theoretical result to compute all its twists. These families of smooth plane curves have already been studied by the first two authors in [3,4]. These families have genus arbitrarily high, so the method in [17] does not work for them.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On twists of smooth plane curves

2018

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Badr

Bars

2016

Mediterr. J. Math.

View full text Add to dashboard Cite

Automorphism Groups of Nonsingular Plane Curves of Degree 5

Cited by 21 publications

References 8 publications

A note on the stratification by automorphisms of smooth plane curves of genus 6

A note on the stratification by automorphisms of smooth plane curves of genus 6

On twists of smooth plane curves

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

Contact Info

Product

Resources

About