On the Xiao conjecture for plane curves

Abstract

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

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

“…Moreover, we prove that if the general fibre is a plane curve of degree ≥ 5 then the stronger bound u f ≤ g − c f − 1 holds. In particular, this provides a strengthening of the bounds of [BGAN15] and of [FPN17]. The strongholds of our arguments are the deformation techniques developed by the first author in [GA16] and by the third author and Pirola in [PT17], which display here naturally their power and depht.…”

“…The inequality (1.5) for fibrations whose general fibres are plane curves uses the results of [FPN17]: as these regard infinitesimal deformations, the very same arguments as above apply to extend Favale-Naranjo-Pirola's inequality to hold for u f .…”

“…(Q6) (Barja-González-Naranjo [BGAN15]) If f is non-isotrivial, then q f ≤ g − c f , where c f is the Clifford index of the general fibre of f . Favale-Naranjo-Pirola [FPN17] have recently proven the stronger inequality q f ≤ g − c f − 1 for families of plane curves of degree d ≥ 5.…”

Fujita decomposition on families of abelian varieties

“…Examples are only known for g ≤ 4 (see [33] and [23]). It has been proved (see [3,8]) that if such a fibration of genus 5 or 6 exists, the fibers of π would be trigonal curves and when g = 6 of special Maroni invariant. If g = 6 using methods from [25,28,30], one could even try to prove that the relative Albanese variety is not simple.…”