Residuation of Linear Series and The Effective Cone of C_d

Abstract

We obtain new information about divisors on the $d-$th symmetric power $C_d$ of a general curve $C$ of genus $g\ge 4.$ 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 $\frac{g+1}{2}\le d\le g-2.$ We also obtain, for each $g\ge 5,$ 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 $2\le d\le g$ and an integral divisor on $C_{g-1}$ which has non-integral volume whenever $g$ is not a power of 2.

19 pages, final version. To appear in American Journal of Mathematics