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.

Abstract. We study the multiplicity of the jumping numbers of an m-primary ideal a in a two-dimensional local ring with a rational singularity. The formula we provide for the multiplicities leads to a very simple and efficient method to detect whether a given rational number is a jumping number. We also give an explicit description of the Poincaré series of multiplier ideals associated to a proving, in particular, that it is a rational function.

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.

Motivated by a conjecture of Xiao, we study supporting divisors of fibred surfaces. On the one hand, after developing a formalism to treat one-dimensional families of varieties of any dimension, we give a structure theorem for fibred surfaces supported on relatively rigid divisors. On the other hand, we study how to produce supporting divisors by constructing a global adjoint map for a fibration over a curve (generalizing the infinitesimal constructions of Collino, Pirola, Rizzi and Zucconi).

We prove the sharp bound of at most 64 lines on projective quartic surfaces S ⊂ P 3 (C) (resp. affine quartics S ⊂ C 3 ) that are not ruled by lines. We study configurations of lines on certain non-K3 surfaces of degree four and give various examples of singular quartics with many lines.