Let \(X\) be a projective complex surface with Picard number \(\rho(X)\). One has the Néron-Severi lattice, i.e., a free \(\mathbb{Z}\)-module of rank \(\rho(X): NS(X)= (H^2(X,\mathbb{Z})/ \text{torsion})\cap H^{1,1} (X,\mathbb{C})\) with the naturally defined integral bilinear form of signature \((1, \rho(X)-1)\) provided by the cup product. Any integral lattice is associated with a finite quadratic form called the discriminant form by V. V. Nikulin [Math. USSR, Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111-177 (1979; Zbl 0408.10011)]. The main result of the paper under review describes, upto isomorphism, elliptic fibrations on a K3 surface \(X\) using its \(NS(X)\) with the associated discriminant form.
More precisely for an elliptic fibration \(\pi:X\to \mathbb{P}^1\) with a canonical section 0, the possible singular fibers were classified by K. Kodaira [Ann. Math., II. Ser. 77, 563-626 (1962; Zbl 0118.15802)]. Let \(\{p_1,\dots, p_n\}\subset \mathbb{P}^1\) be the set of points over which the fibers \(\pi^{-1} (p_i)\) have more than one components. Let \(S_i\) be the lattice generated by the components of \(\pi^{-1}(p_i)\) not touching the 0 section with cup product as the integral bilinear form. Let \(q_i\) be the associated discriminant form. Then the lattice \(S= \bigoplus S_i\) with the discriminant form \(q= \bigoplus q_i\) naturally embeds in \(NS(X)\) as a negative definite sublattice of rank \(\rho(X)-2\) (the other generators of \(NS(X)\) are the 0 section and a fiber). Conversely every elliptic fibration of \(X\) corresponds to, upto an isomorphism, an embedding of some lattice \((S,q)\) in \(NS(X)\) as above with each component \((S_i,q_i)\) (classified in the paper) coming from a singular fiber of some Kodaira type.