  We give bounds on the primes of geometric bad reduction for curves of genus
three of primitive CM type in terms of the CM orders. In the case of genus one,
there are no primes of geometric bad reduction because CM elliptic curves are
CM abelian varieties, which have potential good reduction everywhere. However,
for genus at least two, the curve can have bad reduction at a prime although
the Jacobian has good reduction. Goren and Lauter gave the first bound in the
case of genus two.
  In the cases of hyperelliptic and Picard curves, our results imply bounds on
primes appearing in the denominators of invariants and class polynomials, which
are important for algorithmic construction of curves with given characteristic
polynomials over finite fields.
