On the Bielliptic and Bihyperelliptic Loci

Frediani, Paola; Porru, Paola

doi:10.1307/mmj/1596700820

Cited by 4 publications

(10 citation statements)

References 27 publications

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“…Then, the techniques developed in [10], [8] for the second fundamental form of the Torelli map and in [5], [6] for the second fundamental form of the Prym map, allow to compute these second fundamental forms on the quadrics that we have constructed and show that they do not vanish. A similar technique in the case of cyclic groups, has been used in [19] to show that the bielliptic and the bihyperelliptic loci are not totally geodesic.…”

Section: Ctmentioning

confidence: 99%

“…We endow A g with the Siegel metric, that is the orbifold metric induced on A g by the symmetric metric on the Siegel space H g . The second fundamental form of the Torelli map with respect to the Siegel metric has been studied in [10], [8], [19]. Its dual at a point [C] ∈ M g corresponding to a non hyperelliptic curve C is a map…”

Section: Totally Geodesic Subvarieties In the Torelli Locusmentioning

confidence: 99%

“…The Torelli map restricted to the hyperelliptic locus HE g , j h : HE g → A g , is an orbifold immersion ( [37]) and in [4, Prop. 5.1], [19,Section 6] the second fundamental form of the restriction of the Torelli map to the hyperelliptic locus has been studied.…”

Section: If We Write In Local Coordinates As Abovementioning

confidence: 99%

“…Consider j : M g → A g the Torelli morphism, which is an orbifold immersion outside the hyperelliptic locus ( [37]). The second fundamental form of the Torelli map with respect to the Siegel metric has been studied in [10], [8], [19]. Its dual at a point [C] ∈ M g corresponding to a non hyperelliptic curve C is a map (1.1) ρ : I 2 (K C ) → S 2 H 0 (C, K ⊗2 C ) where I 2 (K C ) is the kernel of the multiplication map S 2 H 0 (C, K C ) → H 0 (C, K 2 C ), that is, the space of quadrics containing the canonical image of C.…”

Section: Introductionmentioning

confidence: 99%

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Abelian covers and second fundamental form

Frediani

2021

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We give some conditions on a family of abelian covers of P 1 of genus g curves, that ensure that the family yields a subvariety of Ag which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if g > M , then the family yields a subvariety of Ag which is not totally geodesic. We prove then analogous results for families of abelian covers of Ct → P 1 = Ct/ G with an abelian Galois group G of even order, proving that under some conditions, if σ ∈ G is an involution, the family of Pryms associated with the covers Ct → Ct = Ct/ σ yields a subvariety of A δ p which is not totally geodesic. As a consequence, we show that if G = (Z/N Z) m with N even, and σ is an involution in G, there exists an integer M (N ) which only depends on N such that, if g = g( Ct) > M (N ), then the subvariety of the Prym locus in A δ p induced by any such family is not totally geodesic (hence it is not Shimura).

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Section: Ctmentioning

confidence: 99%

Section: Totally Geodesic Subvarieties In the Torelli Locusmentioning

confidence: 99%

Section: If We Write In Local Coordinates As Abovementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Abelian covers and second fundamental form

Frediani

2021

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“…(See [4,32] and [29] for a thorough survey. See [5,6,7,9,10,11,12,14,17,15,19,23,24,28] for related results.) Shimura subvarieties are totally geodesic, i.e they are images of totally geodesic submanifolds of the Siegel space, that we denote by S g .…”

Section: Introductionmentioning

confidence: 99%

Infinitely many Shimura varieties in the Jacobian locus for $g \leq 4$

Frediani¹,

Ghigi²,

Spelta³

2019

Preprint

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We study families of Galois covers of curves of positive genus. It is known that under a numerical condition these families yield Shimura subvarieties generically contained in the Jacobian locus. We prove that there are only 6 families satisfying this condition, all of them in genus 2,3 or 4. We also show that these families admit two fibrations in totally geodesic subvarieties, generalizing a result of Grushevsky and Möller. Countably many of these fibres are Shimura. Thus the Jacobian locus contains infinitely many Shimura subvarieties of positive dimension of any g ≤ 4.

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Abelian covers and the second fundamental form

Frediani

2024

manuscripta math.

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We give some conditions on a family of abelian covers of $${\mathbb P}^1$$ P 1 of genus g curves, that ensure that the family yields a subvariety of $${\mathsf A}_g$$ A g which is not totally geodesic, hence it is not Shimura. As a consequence, we show that for any abelian group G, there exists an integer M which only depends on G such that if $$g >M$$ g > M , then the family yields a subvariety of $${\mathsf A}_g$$ A g which is not totally geodesic. We prove then analogous results for families of abelian covers of $${\tilde{C}}_t \rightarrow {\mathbb P}^1 = {\tilde{C}}_t/{\tilde{G}}$$ C ~ t → P 1 = C ~ t / G ~ with an abelian Galois group $${\tilde{G}}$$ G ~ of even order, proving that under some conditions, if $$\sigma \in {\tilde{G}}$$ σ ∈ G ~ is an involution, the family of Pryms associated with the covers $${\tilde{C}}_t \rightarrow C_t= {\tilde{C}}_t/\langle \sigma \rangle $$ C ~ t → C t = C ~ t / ⟨ σ ⟩ yields a subvariety of $${\mathsf A}_{p}^{\delta }$$ A p δ which is not totally geodesic. As a consequence, we show that if $${\tilde{G}}=(\mathbb Z/N\mathbb Z)^m$$ G ~ = ( Z / N Z ) m with N even, and $$\sigma $$ σ is an involution in $${\tilde{G}}$$ G ~ , there exists an integer M(N) which only depends on N such that, if $${\tilde{g}}= g({\tilde{C}}_t) > M(N)$$ g ~ = g ( C ~ t ) > M ( N ) , then the subvariety of the Prym locus in $${{\mathsf A}}^{\delta }_{p}$$ A p δ induced by any such family is not totally geodesic (hence it is not Shimura).

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On the Bielliptic and Bihyperelliptic Loci

Cited by 4 publications

References 27 publications

Abelian covers and second fundamental form

Abelian covers and second fundamental form

Infinitely many Shimura varieties in the Jacobian locus for $g \leq 4$

Abelian covers and the second fundamental form

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