On the Nẹ́ron-severi lattice of Delsarte surfaces

Abstract

The Nẹ́ron-Severi group, NS(X), of a given (non-singular projective) variety, X, is defined in only algebro-geometric terms, however it is also known to be an arithmetic invariant. So it is an important study that helps understanding the geometry of the variety. However, there is no known method to compute it in general. For this reason, one first computes the Picard number p(X) = rnk NS(X) of the variety. There has been many studies which elevated the understanding of p(X) in special cases. Yet the difficulty of the computation in the general case still remains. On the other hand, in the case of Delsarte surfaces, an explicit algorithm to compute p(X) is given by Shioda [1], and Degtyarev [2] showed that a generating set for the Nẹ́ron-Severi group, NS(X) can be computed in some cases. Moreover, Heijne [3] gives a classification of all Delsarte surfaces with only isolated ADE singularities. We give an introduction to Delsarte surfaces, and determine which of the Delsarte surfaces given in [3] fit in the descriptions given in [2].