  Given a curve $C$ over a field $K$, the period of $C/K$ is the gcd of degrees
of $K$-rational divisor classes, while the index is the gcd of degrees of
$K$-rational divisors. S. Lichtenbaum showed that the period and index must
satisfy certain divisibility conditions. For given admissible period, index,
and genus, we show that there exists a curve $C$ and a number field $K$ with
these desired invariants, as long as the index is not divisible by $4$.
