On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces
The main object of the paper under review is the Mordell-Weil lattice (MWL) of a surface \(X\) admitting a (relatively minimal) fibration \(f\colon X\to C\) into curves of genus \(g\geq 1\) over an algebraic curve \(C\) defined over an algebraically closed field \(k\). By definition [see, for example, \textit{T.~Shioda}, Proc. Japan Acad., Ser. A 68, 247-250 (1992; Zbl 0788.14021)], MWL is the group \(J(K)\) (modulo torsion) viewed together with the height pairing (here \(K=k(C)\) and \(J\) stands for the Jacobian of the generic fibre of \(f\)). In the case where \(X\) is a rational elliptic surface, MWL turned out to be closely related to the root lattice \(E_8\). The author considers a natural generalization of this phenomenon. Namely, for \(m\geq 2\) let \[ \Gamma _{4m}=\left\{\sum_{i=1}^{4m}x_ie_i : x_i\in 2\mathbb Z,\;x_1\equiv\dots\equiv x_{4m}\pmod{\mathbb Z},\;\sum_{i=1}^{4m}x_i\equiv 0\pmod{2\mathbb Z}\right\}. \] This is a unimodular lattice in \(\mathbb R^{4m}\) which can be viewed as a natural generalization of the root lattice \(E_8\). (Another common notation for \(\Gamma _{4m}\) is \(D_{4m}^+\), the unique lattice contained strictly between \(D_{4m}\) and its dual.) Such lattices appeared as the MWL for higher genus fibrations on a rational surface over \(\mathbb C\) [see \textit{M.-H.~Saito} and \textit{K.-I.~Sakakibara}, J. Math. Kyoto Univ. 34, 859-871 (1994; Zbl 0860.14034) and \textit{K.-V.~Nguyen}, Osaka J. Math. 34, 101-114 (1997; Zbl 0891.14008)]. The first main result of the paper under review is a generalization of this observation to the case of an arbitrary ground field \(k\) of characteristic \(\neq 2\); in particular, it says that for \(g\geq 2\) a surface \(X\) with maximal possible Mordell-Weil rank is a rational surface admitting a hyperelliptic fibration (in other words, \(X\) can be obtained as a blow-up of a linear pencil of hyperelliptic curves on a Hirzebruch surface \(\Sigma _e\) with \(0\leq e\leq g\)), and its MWL is isometric to \(\Gamma _{4g+4}\). The next result gives another realization of \(\Gamma _{4g+4}\). Namely, let \(NS(X)\) be the Néron-Severi lattice of \(X\), and let \(U\) denote the sublattice of rank 2 generated by the class of a section and the class of the generic fibre. The orthogonal complement to \(U\) in \(NS(X)\) is a negative definite lattice \(W\). The lattice opposite to \(W\) is called the frame lattice of \(X\). The author proves that if \(f\colon X\to\mathbb P^1\) is a relatively minimal fibration into curves of genus \(g\geq 1\) on a rational surface \(X\), whose relative canonical class is zero, then (under a certain additional assumption) the frame lattice of \(X\) is isometric to \(\Gamma _{4g+4}\). The next part of the paper is devoted to a detailed description of the structure of high rank (\(r\geq 4g+1\)) sublattices of the MWL. The classification, building upon case-by-case consideration of the structure of the degenerate fibres, gives analogues of the root lattices \(E_7\), \(E_6\), \(D_5\), \(A_4\), and \(A_2\oplus A_1\) embedded into \(E_8\). As a corollary, the author obtains a generalization of a result of \textit{K.~Oguiso} and \textit{T.~Shioda} [Comment. Math. Univ. St. Pauli 40, 83-99 (1991; Zbl 0757.14011)]: the Mordell--Weil group \(J(K)\) of the Jacobian of the generic fibre of rank \(r\geq 4g+1\) is torsion-free. Finally, the author presents two examples containing detailed analysis of constructions due to \textit{M.-H.~Saito} and \textit{A. Beauville} (semistable fibrations with five singular fibres of genus 2 and 3, respectively); in particular, he computes the Mordell-Weil rank and the torsion part of \(J(K)\) and exhibits free generators for \(J(K)/J(K)_{\text{{tor}}}\) and generators for \(J(K)_{\text{{tor}}}\).
scientific article; zbMATH DE number 1551884
scientific article