Subvarieties of the Moduli Space of Curves Parametrizing Jacobians with Non-Trivial Endomorphisms

Ciliberto, Ciro; Geer, Gerard van der

doi:10.2307/2374769

Cited by 8 publications

(12 citation statements)

References 3 publications

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“…However, Oort tentatively believes Coleman's conjecture may be true when g is sufficiently large. Partial results on Oort's question, which are complementary to results in this paper, have been obtained by Ciliberto, van der Geer and Teixidor i Bigas in [3] and [4].…”

Section: Introductionsupporting

confidence: 77%

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Locally Symmetric Families of Curves and Jacobians

Hain¹

1999

Moduli of Curves and Abelian Varieties

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Section: Introductionsupporting

confidence: 77%

“…One can find further restrictions on the largest strata of Z dec , but so far I have not been able to find a use for them. Partial results in this direction have been obtained in [3] and [4].…”

Section: Geometry Of the Jacobian Locusmentioning

confidence: 99%

Locally Symmetric Families of Curves and Jacobians

Hain¹

1999

Moduli of Curves and Abelian Varieties

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“…Note that r = τ (2, g) − 1 = 2g − 7. The automorphisms group of F a has dimension a + 5 if a > 0 and 6 if a = 0, which explains the first two lines of (10).…”

Section: Castelnuovo Pairsmentioning

confidence: 97%

“…Proof. By Remark 1.1, we may assume X has negative Kodaira dimension, otherwise r ≤ g. If q > 0, there is a curve C of genus q and a surjective morphism f : X → C, hence all curves in |D| map to C and therefore µ(X, D) ≤ 2g − 2 (see [10]). So we may assume q = 0.…”

Section: Number Of Moduli (Ii)mentioning

confidence: 99%

On a theorem of Castelnuovo and applications to moduli

Castorena¹,

Ciliberto²

2011

Kyoto J. Math.

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In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo's theorem. Finally we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.Comment: 8 page

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“…It would be interesting to investigate the structure of such loci. In the case d = 1 these loci have been studied for example in [38,15,11,10].…”

Section: Introductionmentioning

confidence: 99%

Fujita Decomposition and Hodge Loci

Frediani¹,

Ghigi²,

Pirola³

2018

J. Inst. Math. Jussieu

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This paper contains two results on Hodge loci in Mg. The first concerns fibrations over curves with a non-trivial flat part in the Fujita decomposition. If local Torelli theorem holds for the fibres and the fibration is non-trivial, an appropriate exterior power of the cohomology of the fiber admits a Hodge substructure. In the case of curves it follows that the moduli image of the fiber is contained in a proper Hodge locus. The second result deals with divisors in Mg. It is proved that the image under the period map of a divisor in Mg is not contained in a proper totally geodesic subvariety of Ag. It follows that a Hodge locus in Mg has codimension at least 2.In particular we have the following corollary:Corollary 1.4 (See Corollary 5.14). If g ≥ 3, any Hodge locus of M g has codimension at least 2.The proof of Theorem 1.3 is based on a result of independent interest (Theorem 5.8) that describes the behaviour of a divisor in M g at the boundary. This result is a variation on an argument in [29]. It allows to use induction on g. The case g = 3 follows, as a very special case, from a theorem by Berndt and Olmos [3] on the codimension of totally geodesic submanifolds in symmetric spaces, see 5.10. The inductive step depends on simple Lie theoretic computations showing that H k × H g−k is a maximal totally geodesic submanifold of H g , see Proposition 5.6.Acknowledgements. The authors wish to thank Professor Atsushi Ikeda for introducing them to paper [45]. The second author would like to thank Professor J.S. Milne for interesting information on arithmetic varieties.

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Subvarieties of the Moduli Space of Curves Parametrizing Jacobians with Non-Trivial Endomorphisms

Cited by 8 publications

References 3 publications

Locally Symmetric Families of Curves and Jacobians

Locally Symmetric Families of Curves and Jacobians

On a theorem of Castelnuovo and applications to moduli

Fujita Decomposition and Hodge Loci

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