  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.
