Let $X$ be a Riemann Surface with genus $g$, $S^g(X)$ be the symmetric power of $X$ (which is naturally identified with the set of effective divisors of degree $g$). Let $A$ be the Abel-Jacobi map from $S^g(X)$ to $Jac(X)$. We know that for any $D \in S^g(X)$, $A^{-1}A(D)$ is given by the linear system $|D|$.

By computing the rank of $A$, we know that $A$ has constant rank along $A^{-1}A(D)$, but does this (maybe also using the Rank Theorem for holomorphic maps) imply $A^{-1}A(D)$ is a submanifold of $S^g(X)$?