We obtain new information about divisors on the $d-$th symmetric power $C_{ d}$ of a general curve $C$ of genus $g \geq 4.$ This includes a complete description of the effective cone of $C_{ g-1}$ and a partial computation of the volume function on one of its non-nef subcones, as well as new bounds for the effective and movable cones of $C_{ d}$ in the range $\frac{ g+1}{ 2} \leq d \leq g-2.$ We also obtain, for each $g \geq 5,$ a divisor on $C_{ g-1}$ with non-equidimensional stable base locus. For a general hyperelliptic curve $C$ of genus $g,$ we obtain a complete description of the effective cone of $C_{ d}$ for $2 \leq d \leq g$ and an integral divisor on $C_{ g-1}$ which has non-integral volume whenever $g$ is not a power of 2.
Categories: math.AG