Xiao’s conjecture for general fibred surfaces

Barja, Miguel A.; González-Alonso, Víctor; Naranjo, Juan Carlos

doi:10.1515/crelle-2015-0080

Cited by 15 publications

(29 citation statements)

References 18 publications

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“…The results of [BGAN15] imply the conjecture in the (general) case of maximal Clifford index c f = g−1 2 . All counterexamples to the original conjecture found by Albano and Pirola satisfy equality for the modified one.…”

Section: Motivation and Statement Of The Resultssupporting

confidence: 64%

“…Let us spend a couple of words about the proof of Theorem 1.3. 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: 98%

“…Xiao's original conjecture [Xia87b] that the first bound of (Q3) holds for any nontrivial fibration has been disproved by Pirola in [Pir92] (more counterexamples have been found later by Albano and Pirola in [AP16]) and was modified in [BGAN15] as follows.…”

Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

“…We recall now some definitions and results concerning infinitesimal deformations of curves already introduced in [CP95, GA16,BGAN15]. Let C be a smooth curve and ξ ∈ H 1 (C, T C ) the Kodaira-Spencer class of a first-order infinitesimal deformation C ֒→ C, which corresponds to the exact sequence of vector bundles on C…”

Section: Supporting Divisorsmentioning

confidence: 99%

“…The rank of ξ is the most important numerical invariant for our purposes, and is related to supporting divisors by the following Theorem 2.9 (Lemma 2.3 and Thm 2.4 in [BGAN15]). Suppose ξ is supported on D. Then H 0 (C, ω C (−D)) ⊆ ker ∂ ξ .…”

Section: Supporting Divisorsmentioning

confidence: 99%

See 4 more Smart Citations

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 Resultssupporting

confidence: 64%

Section: Motivation and Statement Of The Resultsmentioning

confidence: 98%

Section: Motivation and Statement Of The Resultsmentioning

confidence: 99%

Section: Supporting Divisorsmentioning

confidence: 99%

Section: Supporting Divisorsmentioning

confidence: 99%

See 3 more Smart Citations

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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Infinitesimal Variation Functions for Families of Smooth Varieties

Favale

Pirola

2022

Milan J. Math.

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In this paper we introduce some variation functions associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if X is a smooth plane curve, then, there exists a first order deformation $$\xi \in H^1(T_X)$$ ξ ∈ H 1 ( T X ) which deforms X as plane curve and such that $$\xi \cdot :H^0(\omega _X)\rightarrow H^1({\mathcal O}_{X})$$ ξ · : H 0 ( ω X ) → H 1 ( O X ) is an isomorphism. We also generalize the notions of variation functions to higher dimensional case and we analyze the link between IVHS and the weak and strong Lefschetz properties of the Jacobian ring of a smooth hypersurface.

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On the Xiao conjecture for plane curves

2017

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Let $f: S\longrightarrow B$ be a non-trivial fibration from a complex projective smooth surface $S$ to a smooth curve $B$ of genus $b$. Let $c_f$ the Clifford index of the generic fibre $F$ of $f$. In [arXiv:1401.7502v4] it is proved that the relative irregularity of $f$, $q_f=h^{1,0}(S)-b$ is less than or equal to $g(F)-c_f$. In particular this proves the (modified) Xiao's conjecture: $q_f\le 1+g(F)/2$ for fibrations of general Clifford index. In this short note we assume that the generic fiber of $f$ is a plane curve of degree $d\ge 5$ and we prove that $q_f\le g(F)-c_f-1$. In particular we obtain the conjecture for families of quintic plane curves. This theorem is implied for the following result on infinitesimal deformations: let $F$ a smooth plane curve of degree $d\ge 5$ and let $\xi$ be an infinitesimal deformation of $F$ preserving the planarity of the curve. Then the rank of the cup-product map $\cdot \xi: H^0(F,\omega_F) \rightarrow H^1(F,O_F)$ is at least $d-3$. We also show that this bound is sharp.Comment: 8 pages. Some typos have been corrected. To appear in "Geometriae Dedicata

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Xiao’s conjecture for general fibred surfaces

Abstract: We prove that the genus g, the relative irregularity q f and the Clifford index c f of a non-isotrivial fibration f satisfy the inequality q f ≤ g − c f . This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.

Cited by 15 publications

References 18 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

Infinitesimal Variation Functions for Families of Smooth Varieties

On the Xiao conjecture for plane curves

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