On Mordell-Weil lattices of bielliptic fibrations on rational surfaces (Q1775416)

Let \(X\) be a smooth projective complex surface and \(f: X \to \mathbb{P}^{1}\) be a relatively minimal fibration of curves of genus \(g \geq 1\), in the sense that no fibres contain a \((-1)\)-curve. Assume that the fibration \(f: X \to \mathbb{P}^{1}\) has a section, and let \(K\) be the rational function field of \(\mathbb{P}^{1}\). The Mordell-Weil group of \(f\) is the group \(J_{F}(K)\) of \(K\)-rational points of the Jacobian variety \(J_{F}\) of the general fibre \(F\) of \(f\). Furthermore, assume that \(X\) is a rational surface, then \(J_{F}(K)\) is finitely generated. It is a result of Shioda that \(J_{F}(K) \cong \text{NS}(X)/T\), with \(\text{NS}(X)\) the Néron-Severi group of \(X\) and \(T\) the subgroup generated by the zero section of \(J_{F}(K)\) and all irreducible components of fibres of \(f\). Via the above isomorphism, one obtains a symmetric bilinear form on \(J_{F}(K)\) which induces the structure of a positive-definite lattice on \(J_{F}(K)/J_{F}(K)_{\text{tor}}\). The lattice \(J_{F}(K)/J_{F}(K)_{\text{tor}}\) is called the Mordell-Weil lattice of the fibration \(f: X \to \mathbb{P}^{1}\). In the paper under review, the author deals with the case when \(X\) is rational and the fibration \(f: X \to \mathbb{P}^{1}\) is relatively minimal bi-elliptic of genus \(g \geq 6\), where bi-elliptic means that the general fibre \(F\) of \(f\) is a double cover of an elliptic curve. Under the above conditions, the author proves the following inequality for the rank \(r\) of the group \(J_{F}(K)\): \[ r \leq 2g+10. \] Moreover, the equality \(r=2g+10\) holds if and only if \(K_{X}^{2}=-2g-2\) and all fibres of \(f\) are irreducible. This result is a consequence of a slope inequality. By a refinement of the slope inequality under the above situation, the equality \(r=2g+10\) implies that one has a finite double cover from the surface \(X\) to a smooth rational minimal elliptic surface. An analysis of this double cover gives the result. When the rank of \(J_{F}(K)\) attains the maximum, the author produces explicit models for the fibrations, and he determines the structure of their Mordell-Weil lattices.