On Ballico-Hefez curves and associated supersingular surfaces

Hoang, Thanh Hoai; Shimada, Ichio

doi:10.2996/kmj/1426684441

Cited by 4 publications

(6 citation statements)

References 17 publications

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“…Let G and g be the defining polynomials of C ∨ 0 and B, respectively. Using Proposition 1.6 of [5], if p = 2, then…”

Section: Fermat Curvementioning

confidence: 99%

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Singularities of the dual curve of a certain plane curve in positive characteristic

Komeda¹

2020

Preprint

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It is well known that the Gauss map for a complex plane curve is birational, whereas the Gauss map in positive characteristic is not always birational. Let q be a power of a prime integer. We study a certain plane curve C of degree q 2 + q + 1 for which the Gauss map is inseparable with inseparable degree q. As a special case, we show a relation between the dual curve of the Fermat curve of degree q 2 + q + 1 and the Ballico-Hefez curve.

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“…Let G and g be the defining polynomials of C ∨ 0 and B, respectively. Using Proposition 1.6 of [5], if p = 2, then…”

Section: Fermat Curvementioning

confidence: 99%

“…In [5], Hoang and Shimada define the Ballico-Hefez curve to be the image of the morphism P 1 → P 2 defined by [s : t] → [s q+1 : t q+1 : st q + s q t].…”

Section: Introductionmentioning

confidence: 99%

Singularities of the dual curve of a certain plane curve in positive characteristic

Komeda¹

2020

Preprint

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“…Therefore R is the projective line "with q(q−1) 2 ordinary nodes", i.e., it is P 1 F q where the pairs of points {{a, a q }|a ∈ F q 2 \ F q } are identified. The curve R is known as the Ballico-Hefez curve (see [13] and [16]). Below (7.4 for p = 2 and §7.1.2 for p = 2) we give an equation for this curve (see also [16] prop.…”

Section: An Algebraic Description Of the Family Of Curves X λ For P >mentioning

confidence: 99%

“…The curve R is known as the Ballico-Hefez curve (see [13] and [16]). Below (7.4 for p = 2 and §7.1.2 for p = 2) we give an equation for this curve (see also [16] prop. 1.4).…”

Section: An Algebraic Description Of the Family Of Curves X λ For P >mentioning

confidence: 99%

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Mumford curves and Mumford groups in positive characteristic

Voskuil

Put

2019

Journal of Algebra

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A Mumford group is a discontinuous subgroup Γ of PGL 2 (K), where K denotes a non archimedean valued field, such that the quotient by Γ is a curve of genus 0. As abstract group Γ is an amalgam of a finite tree of finite groups. For K of positive characteristic the large collection of amalgams having two or three branch points is classified. Using these data Mumford curves with a large group of automorphisms are discovered. A long combinatorial proof, involving the classification of the finite simple groups, is needed for establishing an upper bound for the order of the group of automorphisms of a Mumford curve. Orbifolds in the category of rigid spaces are introduced. For the projective line the relations with Mumford groups and singular stratified bundles are studied. This paper is a sequel to [P-V]. Part of it clarifies, corrects and extends work of G. Cornelissen, F. Kato and K. Kontogeorgis. 1 ated, discontinuous subgroup of PGL 2 (K) such that ∆ contains no elements ( = 1) of finite order and ∆ ∼ = {1}, Z. It turns out that ∆ is a free non-abelian group on g > 1 generators. Let Ω ⊂ P 1 K denote the rigid open subspace of ordinary points for ∆. Then X := Ω/∆ is an algebraic curve over K of genus g. The curves obtained in this way are called Mumford curves. Let Γ ⊂ PGL 2 (K) denote the normalizer of ∆. Then Γ/∆ acts on X and is in fact the group of the automorphisms of X. For K ⊃ Q p , the theme of automorphisms of Mumford curves is of interest for p-adic orbifolds and for p-adic hypergeometric differential equations. According to F. Herrlich [He] one has for Mumford curves X of genus g > 1 the bound |Aut(X)| ≤ 12(g − 1) if p > 5. For p = 2, 3, 5 there are p-adic "triangle groups" and the bounds are n p (g − 1) with n 2 = 48, n 3 = 24, n 5 = 30.In this paper we investigate the case that K has characteristic p > 0. The order of the automorphism group can be much larger than 12(g − 1). Using the Riemann-Hurwitz-Zeuthen formula one easily shows (see also the proof of Corollary 6.2):If g > 1 and |Aut(X)| > 12(g − 1), then X/Aut(X) ∼ = P 1 K and the morphism X → X/Aut(X) is branched above 2 or 3 points.There exist Mumford curves X = Ω/∆ with genus g > 1 and such that |Aut(X)| > 12(g − 1). Hence the normalizer Γ of ∆ ⊂ PGL(2, K) satisfies Ω/Γ ∼ = P 1 K . This leads to the definition of a Mumford group: This is a finitely generated, discontinuous subgroup Γ of PGL 2 (K) such that Ω/Γ ∼ = P 1 K , where Ω ⊂ P 1 K is the rigid open subset of the ordinary points for the group Γ. We exclude the possibilities that Γ is finite and that Γ contains a subgroup of finite index, isomorphic to Z. A point a ∈ P 1 K is called a branch point if a preimage b ∈ Ω of a has a non trivial stabilizer in Γ.On the other hand, a Mumford group Γ contains a normal subgroup ∆, which is of finite index and has no elements = 1 of finite order. Thus ∆ is a Schottky group, X := Ω/∆ is a Mumford curve. Above we have excluded the cases that the genus of X is 0 or 1. The group A := Γ/∆ is a subgroup of Aut(X) such that X/A ∼ = P 1 K .In several papers...

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Singularities of the dual curve of a certain plane curve in positive characteristic

Komeda

2021

Kodai Math. J.

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On Ballico-Hefez curves and associated supersingular surfaces

Cited by 4 publications

References 17 publications

Singularities of the dual curve of a certain plane curve in positive characteristic

Singularities of the dual curve of a certain plane curve in positive characteristic

Mumford curves and Mumford groups in positive characteristic

Singularities of the dual curve of a certain plane curve in positive characteristic

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