Given a complex elliptic K3 surface $\pi\colon X\rightarrow \mathbb P^1$, its *discriminant locus* is the divisor $$D = \sum_{i = 1}^s n_i P_i$$ on $\mathbb P^1$ such that $n_i$ is equal to the Euler-Poincaré characteristic of the fiber $\pi^{-1}(P_i)$, where the sum runs over the points $P_i \in \mathbb P^1$ such that $\pi^{-1}(P_i)$ is singular. It is well known that $\deg D = 24$.

Conversely, given an effective divisor $D$ of degree $24$ on $\mathbb P^1$, when is it the discriminant locus of a complex elliptic K3 surface?

I am particularly curious about the minimal possible $s$. The maximal Euler-Poincaré characteristic of a singular fiber is $20$, so $s \geq 2$. But in case, say, $n_1 = 20$, then the fibration is of type $I_{14}^*,I_1,I_1,I_1,I_1$ (see Schütt-Schweizer), so indeed $s = 5$. Are smaller $s$ possible?

elliptic curvesis hyperbolic. (Although the moduli space of polarized K3 surfaces is hyperbolic as well, it is the moduli space of elliptic curves which plays a role here.) Indeed, if $D$ is the support of the discriminant divisor of the elliptic surface $f:X\to \mathbb{P}^1$, then there is a non-constant morphism $\mathbb{P}^1 \setminus D \to \mathcal{M}$ induced by the Jacobian of $X\setminus f^{-1}D\to \mathbb{P}^1\setminus D$, where $\mathcal{M}$ is the moduli of elliptic curves. $\endgroup$ – Ariyan Javanpeykar Oct 8 '18 at 14:46