Complex surfaces polarized by an ample and spanned line bundle of genus three
Let X be a nonsingular projective surface and L an ample line bundle on X. Its sectional genus \(g\) is defined by the formula \(2g-2=L\cdot (K\otimes L)\), where K is the canonical bundle on X. It is further assumed that L is spanned by its global sections. In the paper under review the authors classify all such pairs (X,L) with \(g=3\) (The cases \(g\leq 2\) are known previously). These are: (1) a double plane, (2) elliptic surfaces with multiple fibres and (3) ruled surfaces, and the third case is further classified. As remarked at the end of the paper, the same classification was independently carried out by \textit{H. Maeda} [Sci. Pap. Coll. Arts. Sci., Univ. Tokyo 37, No.2, 103-112 (1988)] without assuming that L is spanned by its global sections.
scientific article; zbMATH DE number 4160934