Assume we work over $\mathbb{C}$.

Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ such that $S\cong A/\{\pm 1\}$ given by the linear system $|2\theta|: A \rightarrow \mathbb{P}^3$ and the singularities are exactly the images of the 2-torsion points.

  The Torelli theorem states that there is a unique smooth projective curve $C$ of genus $g=2$ such that $(A,\theta)=(Jac(C),C)$.

$\textbf{Question 1:}$ Let $S$ and $S'$ be two quartic surfaces with 16 nodes in $\mathbb{P}^3$ with corresponding curves $C$ and $C'$ of genus 2. If $S$ and $S'$ are birational, is there any relationship between $C$ and $C'$?

(We may assume that the birational map $\phi: S --> S'$ is induced by a birational map $\Psi: \mathbb{P}^3 -->\mathbb{P}^3$.)

My thoughts on this: assume the Picard numbers of $S$ and $S'$ are one, that is the associated abelian surfaces also have Picard number one. Since $S$ and $S'$ are birational, so are their minimal resolutions (blowing up the 16 nodes) $X$ and $X'$ which are $K3$-surfaces. So they are isomorphic. By a result of Inose this implies $A\cong A'$ as abelian surfaces, because they both have Picard number one and a principal polarization. But the pullback of this isomorphism must map $\theta'$ to $\theta$ so that it is in fact an isomorphism of principally polarized abelian surfaces $(A,\theta) \cong (A',\theta')$ which implies $C\cong C'$.

$\textbf{Questions 2:}$ If this result above is the best we can get, is there an "easy" extra condition for example on the birational map $\phi$ such that we can conclude: $S$ and $S'$ are birational $\Rightarrow$ $C\cong C'$?

Assume we work over $\mathbb{C}$.

Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ such that $S\cong A/\{\pm 1\}$ given by the linear system $|2\theta|: A \rightarrow \mathbb{P}^3$ and the singularities are exactly the images of the 2-torsion points. The Torelli theorem states that there is a unique smooth projective curve $C$ of genus $g=2$ such that $(A,\theta)=(Jac(C),C)$.

Now, let $S_C$ and $S_{C'}$ be two quartic surfaces with 16 nodes in $\mathbb{P}^3$ with corresponding curves $C$ and $C'$ of genus 2. Assume $S_C$ and $S_{C'}$ are birational. This implies that their minimal resolutions (blowing up the 16 nodes) $X_C$ and $X_{C'}$ are also birational. But $X_C$ and $X_{C'}$ are $K3$-surfaces, which means they are already isomorphic.

$\textbf{Question:}$ How much information about a curve $C$ is encoded in the $K3$-surface $X_C$? For example in the setting described above:

Does $X_C\cong X_{C'}$ imply $C\cong C'$?

If $C\cong C'$ is not true not in general, can we relate $C$ and $C'$ in any other way in this setting?

My thoughts on this: assume the Picard numbers of $S_C$ and $S_{C'}$ are one, that is the associated abelian surfaces also have Picard number one. Using $X_C\cong X_{C'}$ we get $A\cong A'$ as abelian surfaces by a result of Inose, because they both have Picard number one and a principal polarization. But the pullback of this isomorphism must map $\theta'$ to $\theta$ so that it is in fact an isomorphism of principally polarized abelian surfaces $(A,\theta) \cong (A',\theta')$ which implies $C\cong C'$.

$\textbf{Question 2:}$ What happens for $\rho(A)>1$? The case $\rho(A)=1$ is the very general case in the moduli space of principally polarized abelian surfaces. Can we get the other curves by some limit or deformation argument maybe?

$\textbf{Question 3:}$ If we assume $X_C\cong X_{C'}$ and $J(C)\cong J(C')$ (as abelian surfaces) is this enough to see that $C\cong C'$?