Invariant Spin Structures on Riemann Surfaces

Kallel, Sadok; Sjerve, Denis

doi:10.5802/afst.1251

Cited by 7 publications

(14 citation statements)

References 14 publications

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“…The question which automorphisms σ C of C lift to a spin curve (X, L, b) with stable model C is difficult, even in the case of a smooth curve C. By Atiyah's result for every automorphism σ C ∈ Aut C of a smooth curve C there exists at least one theta characteristic L on C such that σ C lifts to the spin curve (C, L, b). Moreover, S. Kallel and D. Sjerve show in [KS06] that an automorphism σ C ∈ Aut C, where C is smooth, lifts to every theta characteristic L on C if and only if C is hyperelliptic and σ C is the hyperelliptic involution. Therefore, for every automorphism σ C which is not a hyerelliptic involution there exists at least one theta characteristic to which σ C lifts and at least one to which it does not lift.…”

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confidence: 99%

On the geometry of the moduli space of spin curves

Ludwig

2009

J. Algebraic Geom.

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Abstract. We determine the smooth locus and the locus of canonical singularities in the Cornalba compactification S g of the moduli space S g of spin curves, i.e., smooth curves of genus g with a theta characteristic. Moreover, the following lifting result for pluricanonical forms is proved: Every pluricanonical form on the smooth locus of S g extends holomorphically to a desingularisation of S g .

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confidence: 99%

On the geometry of the moduli space of spin curves

Ludwig

2009

J. Algebraic Geom.

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“…This theorem was also proved using different methods in [9]. Yet, We prove it here using our methods as in the proof of Theorem 20.…”

Section: Invariant M-spin Structures On a Hyperelliptic Surface Undermentioning

confidence: 81%

“…In other words, f fixes 2 2g+2 n −2 2-spin structures. This theorem was proved using different methods in [9]. We prove it here using similar methods to the ones we used to prove Theorems 16 and 18.…”

Section: Invariant M-spin Structures On a Hyperelliptic Surface Undermentioning

confidence: 98%

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Spin structures and branch divisors on $p$-gonal Riemann surfaces

Almalki¹,

Nolder²

2019

Rocky Mountain J. Math.

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We study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and p-gonal surfaces defined by divisors supported on their branch points. Moreover, we study invariant spin divisors under automorphisms and anti-holomorphic involutions of Riemann surfaces.Let Div(X) and PDiv(X) denote the group of divisors and the group of principal divisors; respectively. The group Jac(X) = Div 0 (X)/ PDiv(X) is called the Jacobian of X. It is well-known that the Jacobian group of a genus g compact Riemann surface can be given a complex structure, namely it is a complex torus of dimension g. For example the Jacobian of a torus is a torus itself, J(T ) = T . The Jacobian also is an abelian variety. A detailed discussion of these facts can be found for example in the book of Miranda [11]. For our purposes, we only need the group

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“…Assume that X admits nontrivial automorphisms; fix a nontrivial automorphism σ of X . In [3] it was shown that if σ fixes all the theta characteristics of X pointwise, then X is hyperelliptic and σ is the unique hyperelliptic involution of X (this was also proved in [5]). Our aim here is to address a similar question for curves defined over the field of real numbers.…”

Section: Introductionmentioning

confidence: 80%

Real Theta Characteristics and Automorphisms of a Real Curve

Biswas

Gadgil

2010

J. Aust. Math. Soc.

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Let X be a geometrically irreducible smooth projective curve defined over R, of genus at least 2, that admits a nontrivial automorphism, σ . Assume that X does not have any real points. Let τ be the antiholomorphic involution of the complexification x C of X . We show that if the action of σ on the set S(X ) of all real theta characteristics of X is trivial, then the order of σ is even, say 2k, and the automorphism τ • σ k of X C has a fixed point, where σ k is the automorphism of X × R C defined by σ . We then show that there exists X with a real point and admitting a nontrivial automorphism σ , such that the action of σ on S(X ) is trivial, while X/ σ = P 1 R . We also give an example of X with no real points and admitting a nontrivial automorphism σ , such that the automorphism τ • σ k has a fixed point, the action of σ on S(X ) is trivial, and X/ σ = P 1 R .2000 Mathematics subject classification: primary 14P25; secondary 14F10. Keywords and phrases: real curve, real theta characteristic, automorphism. IntroductionLet X be a smooth complex projective curve of genus g, with g ≥ 2. Assume that X admits nontrivial automorphisms; fix a nontrivial automorphism σ of X . In [3] it was shown that if σ fixes all the theta characteristics of X pointwise, then X is hyperelliptic and σ is the unique hyperelliptic involution of X (this was also proved in [5]). Our aim here is to address a similar question for curves defined over the field of real numbers. Let X be a geometrically irreducible smooth projective curve defined over R. Letbe the antiholomorphic involution of the Riemann surface X × R C. The genus of X , which will be denoted by g, is assumed to be at least 2. Let Pic g−1 (X ) R denote the real points of the Picard variety Pic g−1 (X ). The real theta characteristics of X are all those points of Pic g−1 (X ) R which are square roots of the real point of Pic 2g−2 (X ) given by

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Invariant Spin Structures on Riemann Surfaces

Cited by 7 publications

References 14 publications

On the geometry of the moduli space of spin curves

On the geometry of the moduli space of spin curves

Spin structures and branch divisors on $p$-gonal Riemann surfaces

Real Theta Characteristics and Automorphisms of a Real Curve

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