On deformations of curves supported on rigid divisors

González-Alonso, Víctor

doi:10.1007/s10231-014-0455-x

Cited by 12 publications

(25 citation statements)

References 16 publications

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“…The proof of the first inequality (1.4) follows the lines of the original argument of [BGAN15], but in this more general setting new results are needed. The key points of our arguments are the deformation thecnhniques developed by the first author in [GA16] and by the third author and Pirola in [PT17], together with an ad-hoc Castelnuovo-de Franchis theorem for tubular surfaces.…”

Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

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On the rank of the flat unitary summand of the Hodge bundle

González-Alonso

Stoppino

Torelli

2019

Trans. Amer. Math. Soc.

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Let f : S → B f\colon S\to B be a nonisotrivial fibered surface. We prove that the genus g g , the rank u f u_f of the unitary summand of the Hodge bundle f ∗ ω f f_*\omega _f , and the Clifford index c f c_f satisfy the inequality u f ≤ g − c f u_f \leq g - c_f . Moreover, we prove that if the general fiber is a plane curve of degree ≥ 5 \geq 5 , then the stronger bound u f ≤ g − c f − 1 u_f \leq g - c_f-1 holds. In particular, this provides a strengthening of bounds proved by M. A. Barja, V. González-Alonso, and J. C. Naranjo and by F. F. Favale, J. C. Naranjo, and G. P. Pirola. The strongholds of our arguments are the deformation techniques developed by the first author and by the third author and G. P. Pirola, which display here naturally their power and depth.

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Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

“…We extend now the previous definitions to the case of one-dimensional families of curves as done in [GA16]. In order to simplify the exposition, we will assume that the family has no singular members, though the theroy can be developed in more general cases.…”

Section: Supporting Divisorsmentioning

confidence: 99%

On the rank of the flat unitary summand of the Hodge bundle

González-Alonso

Stoppino

Torelli

2019

Trans. Amer. Math. Soc.

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“…This bound has appeared recently in the work of Barja, González-Alonso and Naranjo (see [6] and [2]). The main result of [2] says that for a non isotrivial fibration π : S → B one has…”

Section: Introductionmentioning

confidence: 80%

“…Let H g,p be the moduli space of unramified cyclic covers of degree p of hyperelliptic curves of genus g. A point in H g,p is (up to isomorphism) a pair (E, H ′ ) where E is an hyperelliptic curve of genus g and H ′ is a cyclic subgroup of order p of Pic 0 (E). The dihedral construction of diagram (5) determines uniquely the isomorphism class of D, since any two lifts of the hyperelliptic involution are conjugated in Aut(C) and hence gives a morphism (6) ψ :…”

Section: Introductionmentioning

confidence: 99%

Dihedral monodromy and Xiao fibrations

Albano

Pirola

2015

Annali di Matematica

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We construct three new families of fibrations π : S → B where S is an algebraic complex surface and B a curve that violate Xiao's conjecture relating the relative irregularity and the genus of the general fiber. The fibers of π are certain étale cyclic covers of hyperelliptic curves that give coverings of P 1 with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill-Noether range.

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“…references in this topics are[CP95],[GA16],[PZ03],[NPZ04],[Riz08]. For details on deformation theory and variation of the Hodge structure instead we refer to[Gri83],[GH83a],[GH83b],[Voi02] and also[Voi03].…”

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confidence: 99%

Massey Products and Fujita decompositions on fibrations of curves

Pirola

Torelli

2019

Collect. Math.

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Let f : S → B be a fibred surface and f * ω S/B = U ⊕A be the second Fujita decomposition of f. We study a Massey product related with variation of the Hodge structure over flat sections of U. We prove that the vanishing of the Massey product implies that the monodromy of U is finite and described by morphisms over a fixed curve. The main tools are a lifting lemma of flat sections of U to closed holomorphic forms of S and two classical results due (essentially) to de Franchis. As applications we find a new proof of a theorem of Luo and Zuo for hyperelliptic fibrations. We also analyze, as for the surfaces constructed by Catanese and Dettweiler, the case when U has not finite monodromy.Abstract. Let f : S → B be a fibration of curves and let f * ω S/B = U ⊕ A be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of U.The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole U of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).

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On deformations of curves supported on rigid divisors

Cited by 12 publications

References 16 publications

On the rank of the flat unitary summand of the Hodge bundle

On the rank of the flat unitary summand of the Hodge bundle

Dihedral monodromy and Xiao fibrations

Massey Products and Fujita decompositions on fibrations of curves

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