Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the form $\mathcal O_{J(C)}(m \Theta)$?
I think I remember seeing something of this type years ago, but I haven't been able to find anything related on the web. Maybe I just imagined it.