The Picard number of certain algebraic surfaces
Consider the Picard number \(r\) of a nonsingular projective variety \(X\) (i.e., the rank of its Néron-Severi group) and its Lefschetz number (i.e. \(b_ 2-r\)). The author starts by reviewing some results Shioda which can be applied to compute the Picard number of a Fermat hypersurface and the Lefschetz number of a Delsarte surface. Afterwards families \(f:X\to S\) of algebraic varieties are considered; the theory of variation of Hodge structure is used to obtain an upper bound for the Picard number of the fibre \(X_ s\) of \(f\). The computation is made very explicit and effective for some families of algebraic surfaces (among which a family where one fibre is the Fermat surface of degree 5).
scientific article; zbMATH DE number 167303
scientific article