# Cutting out a Riemann surface inside its Jacobian variety

After choosing a base point $$P_{0}$$ in a compact Riemann surface $$X$$ of genus $$g$$, the Abel-Jacobi map gives an embedding of $$X$$ into its Jacobian variety $$Jac(X).$$ This map can also be extended to define a function from the $$m$$-th symmetric power $$S^{m}(X)$$ to $$Jac(X)$$ which, according to Jacobi's inversion theorem, is surjective when $$m\geq g.$$

Riemann was able to identify the image of $$S^{g-1}(X)$$ in $$Jac(X)$$ as a translate of the zeroes of his theta function. Is there a similar description of the image of $$S^{m}(X)$$ for $$1\leq m\leq g-1$$? In particular, is it possible to cut out $$X$$ inside its Jacobian variety using some more or less canonically defined theta functions?