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Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this well-known fact? Is it possible to explicitly construct such a fibration via the Enriques lattice $NS(S)\cong U\oplus E_8$?

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A complete and detailed treatment of elliptic pencils on Enriques surfaces (included the result you are quoting) can be found in the book by Barth-Hulek-Peters-Van de Ven Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer).

See in particular Chapter VIII, Section 17 "Elliptic pencils".

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    $\begingroup$ Even though this is an accepted answer for an old question, where does it say in Compact Complex Surfaces that the elliptic fibration on $S$ has 12 nodal singular fibres? Is it even true (that there always exists an elliptic fibration with this configuration, not just generically)? $\endgroup$
    – dfn
    Feb 3, 2021 at 21:19

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