  We study how Tamagawa numbers of Jacobians of hyperelliptic curves vary as
one varies the base field or the curve, in the case of semistable reduction. We
find that there are strong constraints on the behaviour that appears, some of
which are unexpected and specific to hyperelliptic curves. Our methods are
explicit and allow one to write down formulae for Tamagawa numbers of infinite
families of hyperelliptic curves, of the kind used in proofs of the parity
conjecture for Jacobians of curves of small genus.
