  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).
