This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.

Let $X=J(C)$ be the Jacobian of a genus 2 curve over $\mathbb{C}$, so $X$ is an abelian surface. We can choose an embedding of $C$ in $X$ such that the hyperelliptic involution on $C$ extends to the involution on $X$. Consider $L=\mathcal{O}(nC)$, where $n$ is an even positive integer. Then $L$ is a totally symmetric line bundle. So $L=f^*L'$ where $f:J(C) \longrightarrow K(C)$ is the quotient map from the Jacobian to the associated Kummer surface, and $L'$ is an ample line bundle on $K(C)$.

Consider a general smooth curve $C'\in |L|$ through all sixteen 2-torsion points. Let $C''$ be its image in $K(C)$, this passes through all sixteen singular points of $K(C)$.

a) For a general smooth $C'$ as above, will the restriction of $f$ from $C' \longrightarrow C''$ be a double cover, or an isomorphism or what?

b) For a general smooth $C'$ as above, is the image $C''$ smooth?

c) When is such a $C'$ isomorphic to its image $C''$?

Thanks in advance for the help!

This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.

Let $X=J(C)$ be the Jacobian of a genus 2 curve over $\mathbb{C}$, so $X$ is an abelian surface. We can choose an embedding of $C$ in $X$ such that the hyperelliptic involution on $C$ extends to the involution on $X$. Consider $L=\mathcal{O}(nC)$, where $n$ is an even positive integer. Then $L$ is a totally symmetric line bundle. So $L=f^*L'$ where $f:J(C) \longrightarrow K(C)$ is the quotient map from the Jacobian to the associated Kummer surface, and $L'$ is an ample line bundle on $K(C)$.

Consider a general smooth curve $C'\in |L|$ through all sixteen 2-torsion points. Let $C''$ be its image in $K(C)$, this passes through all sixteen singular points of $K(C)$.

a) For a general smooth $C'$ as above, will the restriction of $f$ from $C' \longrightarrow C''$ be a double cover, or an isomorphism or what?

b) For a general smooth $C'$ as above, is the image $C''$ smooth?

c) When is such a $C'$ isomorphic to its image $C''$?

Thanks in advance for the help!

This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.

Let $X=J(C)$ be the Jacobian of a genus 2 curve over $\mathbb{C}$, so $X$ is an abelian surface. We can choose an embedding of $C$ in $X$ such that the hyperelliptic involution on $C$ extends to the involution on $X$. Consider $L=\mathcal{O}(nC)$, where $n$ is an even positive integer. Then $L$ is a totally symmetric line bundle. So $L=f^*L'$ where $f:J(C) \longrightarrow K(C)$ is the quotient map from the Jacobian to the associated Kummer surface, and $L'$ is an ample line bundle on $K(C)$.

Consider a general smooth curve $C'\in |L|$ through all sixteen 2-torsion points. Let $C''$ be it's image in $K(C)$, this passes through all sixteen singular points of $K(C)$.

a) For a general smooth $C'$ as above, will the restriction of $f$ from $C' \longrightarrow C''$ be a double cover, or an isomorphism or what?

b) For a general smooth $C'$ as above, is the image $C''$ smooth?

c) When is such a $C'$ isomorphic to it's image $C''$?

Thanks in advance for the help!

This is related to a question that I already asked Curve through the 16 singular points of a Kummer surface. I am new to Abelian varieties and I am trying to understand more about them.

Let $X=J(C)$ be the Jacobian of a genus 2 curve over $\mathbb{C}$, so $X$ is an abelian surface. We can choose an embedding of $C$ in $X$ such that the hyperelliptic involution on $C$ extends to the involution on $X$. Consider $L=\mathcal{O}(nC)$, where $n$ is an even positive integer. Then $L$ is a totally symmetric line bundle. So $L=f^*L'$ where $f:J(C) \longrightarrow K(C)$ is the quotient map from the Jacobian to the associated Kummer surface, and $L'$ is an ample line bundle on $K(C)$.

Consider a general smooth curve $C'\in |L|$ through all sixteen 2-torsion points. Let $C''$ be its image in $K(C)$, this passes through all sixteen singular points of $K(C)$.

a) For a general smooth $C'$ as above, will the restriction of $f$ from $C' \longrightarrow C''$ be a double cover, or an isomorphism or what?

b) For a general smooth $C'$ as above, is the image $C''$ smooth?

c) When is such a $C'$ isomorphic to its image $C''$?

Thanks in advance for the help!