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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?

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