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. 



Resolution of the ideal of the Abel-Jacobi image of a curve?
This is not true, as soon as $g\geq 3$. Taking Chern classes this would imply that $c\_{g-1}(\mathcal{O}\_{a(C)})$ is an integral multiple of $\ \Theta ^{g-1}$ in $\ H^{2g-2}(JC,\mathbb{Z})$. But $c\_{g-1}(\mathcal{O}\_{a(C)})=(-1)^{g}(g-2)![a(C)]=(-1)^g\frac{\Theta ^{g-1}}{g-1}\ $, a contradiction. 


