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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.

user190964
  • 185
  • Abel-Jacobi map of a curve is an embedding ,so $u(X)$ is a genus $g$ curve isomorphic to itself. The differential of the Abel-Jacobi map is identified with canonical map of $X$. So $X$ is a hyperelliptic curve, the differential of the Abel-Jacobi map is 2-to-1, while when $X$ is non-hyperelliptic, the differential is an embedding again. – AG learner Dec 25 '21 at 05:31

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