Period and index for higher genus curves

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.