Generators of the Néron-Severi Group of a Fermat Surface

Abstract

The Néron-Severi group of a (nonsingular projective) variety is, by definition, the group of divisors modulo algebraic equivalence, which is known to be a finitely generated abelian group (cf. [2]). Its rank is called the Picard number of the variety. Thus the Néron-Severi group is defined in purely algebro-geometric terms, but it is a rather delicate invariant of arithmetic nature. Perhaps, because of this reason, it usually requires some nontrivial work before one can determine the Picard number of a given variety, let alone the full structure of its Néron-Severi group. This is the case even for algebraic surfaces over the field of complex numbers, where it can be regarded as the subgroup of the cohomology group H ²(X, ℤ) characterized by the Lefschetz criterion.