General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive integers $d, t$. Such K3 surfaces are necessarily elliptic (because they have class of square zero), and conversely any projective elliptic K3 surface of Picard rank two has such a Neron-Severi lattice. Here $t$ is the minimal degree of a multisection (which is uniquely defined) and $2d$ is the degree of polarization (which is only well-defined mod. $t$). Elliptic surfaces with a section correspond to the case $t = 1$: for any $d$ such lattice is isomorphic to the hyperbolic plane $U$.

The moduli space. By the Torelli theorem for K3 surfaces, for fixed $d, t > 0$ such K3 surfaces exist and they form an $18$-dimensional moduli space $\mathcal{M}_{d,t}$. (These elliptic surfaces have either one or two elliptic fibrations depending on $d, t$.)

Singular fibers. Assume that singular fibers of the elliptic fibrations have a single node (type $I_1$) so that by the Euler characteristic count there are $e(X) = 24$ singular fibers.

Question: do these $24$ points determine the general elliptic K3 from $\mathcal{M}_{d,t}$ up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Possibly if one deals with the $\mathcal{M}_{0,1}$ (elliptic surfaces with a section) the general case will follow by considering the Jacobian fibration (as suggested by user25309 in the comments).

Any suggestions or references welcome.

General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive integers $d, t$. Such K3 surfaces are necessarily elliptic (because they have class of square zero), and conversely any projective elliptic K3 surface of Picard rank two has such a Neron-Severi lattice. Here $t$ is the minimal degree of a multisection (which is uniquely defined) and $2d$ is the degree of polarization (which is only well-defined mod. $t$). Elliptic surfaces with a section correspond to the case $t = 1$: for any $d$ such lattice is isomorphic to the hyperbolic plane $U$.

The moduli space. By the Torelli theorem for K3 surfaces, for fixed $d, t > 0$ such K3 surfaces exist and they form an $18$-dimensional moduli space $\mathcal{M}_{d,t}$. (These elliptic surfaces have either one or two elliptic fibrations depending on $d, t$.)

Singular fibers. Assume that singular fibers of the elliptic fibrations have a single node (type $I_1$) so that by the Euler characteristic count there are $e(X) = 24$ singular fibers. These singular fibers are parametrized by a $21$-dimensional variety $W_{24}$ as we consider $24$ distinct unordered points on $\mathbb{P}^1$ with three of them can be fixed to be $0, 1, \infty$.

Question: are there finitely many elliptic K3 from $\mathcal{M}_{d,t}$ with fixed positions of singular fibers? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers? Equivalently I am asking about the fibers of the rational map $\mathcal{M}_{d,t} \to W_{24}$ which takes elliptic fibration to its set of branch points.

Possibly if one deals with the $\mathcal{M}_{0,1}$ (elliptic surfaces with a section) the general case will follow by considering the Jacobian fibration (as suggested by user25309 in the comments); this will need the argument that fibers of taking the Jacobian fibration map $\mathcal{M}_{d,t} \to \mathcal{M}_{0,1}$ are also finite.

Any suggestions or references welcome.

Elliptic K3 surfaces. Let $X$ be a general projective elliptic K3 of Picard rank two. Assume that singular fibers of the elliptic fibrations are of type $A_1$ so that by the Euler characteristic count there are $e(X) = 24$ singular fibers. I do not require existence of a section in the definition of an elliptic K3 surface.

Question: do these $24$ points determine the general such elliptic K3 uniquely, or at least up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Any suggestions or references welcome.

General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive integers $d, t$. Such K3 surfaces are necessarily elliptic (because they have class of square zero), and conversely any projective elliptic K3 surface of Picard rank two has such a Neron-Severi lattice. Here $t$ is the minimal degree of a multisection (which is uniquely defined) and $2d$ is the degree of polarization (which is only well-defined mod. $t$). Elliptic surfaces with a section correspond to the case $t = 1$: for any $d$ such lattice is isomorphic to the hyperbolic plane $U$.

The moduli space. By the Torelli theorem for K3 surfaces, for fixed $d, t > 0$ such K3 surfaces exist and they form an $18$-dimensional moduli space $\mathcal{M}_{d,t}$. (These elliptic surfaces have either one or two elliptic fibrations depending on $d, t$.)

Singular fibers. Assume that singular fibers of the elliptic fibrations have a single node (type $I_1$) so that by the Euler characteristic count there are $e(X) = 24$ singular fibers.

Question: do these $24$ points determine the general elliptic K3 from $\mathcal{M}_{d,t}$ up to finitely many choices? If not, what can be said about the dimension of the moduli space of those elliptic K3 surfaces with a prescribed set of $24$ singular fibers?

Possibly if one deals with the $\mathcal{M}_{0,1}$ (elliptic surfaces with a section) the general case will follow by considering the Jacobian fibration (as suggested by user25309 in the comments).

Any suggestions or references welcome.