Elliptic surfaces with a nef line bundle of genus two
Complex projective surfaces endowed with a nef line bundle of genus \(\leq 1\) have been classified by \textit{H. Maeda} [Kodai Math. J. 18, No. 1, 187-197 (1995; Zbl 0854.14005)]. In this paper elliptic surfaces \(S\) endowed with a nef line bundle \(L\) with genus \(g(L)=2\) are studied. In particular some techniques worked out by \textit{F. Serrano} [Pac. J. Math. 152, No. 1, 187-199 (1992; Zbl 0756.14026)] and \textit{T. Fujita} [``On certain polarized elliptic surfaces'', in: Geometry of complex projective varieties, Proc.\ Cetraro 1990, Sem. Conf. 9, 153-163 (1993)] to study elliptic quasi-bundles polarized by an ample line bundle of genus \(2\) apply to this more general setting. First, up to contracting all \((-1)\)-curves not intersecting \(L\), one can reduce to the case when either \(S\) is minimal or \((S,L)\) admits a simple reduction \((S',L')\), where \(S'\) is minimal and \(L'\) is a nef line bundle on \(S'\) with \({L'}^2 = L' K_{S'} = 1\). Then one can analyze minimal surfaces by means of the canonical bundle formula determining the possible values of numerical invariants: \(p_g\), \(q\), the genus of the base curve and the multiplicities of the multiple fibres. In several cases the surface is a quasi-bundle over a smooth curve \(C\) of genus \(g(C) \leq 1\). In particular, when \(g(C)=0\) a structure theorem for \(\text{Num}(S)\) allows the authors to give an explicit description of the class of \(L\).
scientific article; zbMATH DE number 1186309