Let $X$ be hyperelliptic of genus $g\geq 2$. For any $0< 2k \leq g$, find a degree $2k$ effective divisor $D$ in $X$ such that $\dim |D| = k$. Classify all such divisors up to linear equivalence.
By Clifford theorem, if $X$ is not hypereplliptic we have $D$ is either principal or canonical. But what happens in $X$ is hyperelliptic?
Reference: Algebraic Curves and Riemann Surfaces by Rick Miranda, http://www.its.caltech.edu/~pablos/files/AlgCurv-RS-Miranda.pdf.