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For a $K_3$ surface $X$, if there exists a holomorphic surjective map $X\to \mathbb P^1$, with elliptic fibres, i.e. for any generic point on $\mathbb P^1$ whose fiber is diffeomorphic to a torus $\mathbb T^2$.

Q: Where we can find the classification of the singularities(the fiber over some point is not a standard manifold, $\mathbb T^2$) of the map $X\to \mathbb P^1$?

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    $\begingroup$ This is a very classical question, which has been answered by Kodaira more than 50 years ago. You'll find an account for instance in Compact complex surfaces by Barth-Hulek-Peters-Van de Ven. $\endgroup$
    – abx
    Dec 20, 2016 at 11:15
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    $\begingroup$ By the way, in algebraic geometry elliptic singularity has a different meaning than singular fibre of an elliptic fibration. $\endgroup$
    – Francesco Polizzi
    Dec 20, 2016 at 12:23

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