For a fixed point $x_0\in X$ of a hyperelliptic curve(genus $g$),
we can think of the image of Abel-Jacobi map $u: x\mapsto (\int_{x_0}^{x}\omega_1,\ldots,\int_{x_0}^{x}\omega_g)$ into its Jacobian $J(X)$.
Then what is it look like?
(1) I think its image should be a curve in $J(X)$. Is it right?
(2) Also, if its image is a curve, then is this curve of genus $g$?
Since I know very little about this theory, it is hard to get an answer... and it is hard to search an answer as well.
Thanks in advance.