A Generalized Torelli Theorem

Given a smooth projective curve $C$ of genus $g$ over the complex numbers, Torelli's thoerem asserts that the pair $(J(C),W^{g-1})$ determines $C$, where $W^{g-1}$ is an image of the $g-1$st symmetric power of $C$ inside the Jacobian under an Abel-Jacobi map. We show that the theorem holds with $g-1$ replaced by an integer $d$ in the range $1\le d\le g-1$.