We construct examples of non-isotrivial algebraic families of smooth
complex projective curves over a curve of genus 2. This solves a problem
from Kirby's list of problems in low-dimensional topology. Namely, we show
that 2 is the smallest possible base genus that can occur in a 4-manifold
of non-zero signature which is an oriented fiber bundle over a Riemann
surface. A refined version of the problem asks for the minimal base genus
for fixed signature and fiber genus. Our constructions also provide new
(asymptotic) upper bounds for these numbers.