Weighted surfaces with maximal Picard number

Abstract

An algorithm due to Shioda computes the Picard number for certain surfaces which are defined by a single equation with exactly four monomials, called Delsarte surfaces. We consider this method for surfaces in weighted projective $3$-space with quotient singularities. We give a criterion for such a weighted Delsarte surface $X$ to have maximal Picard number. This condition is surprisingly related to the automorphism group of $X$. For every positive integer $s$, we find a weighted Delsarte surface with geometric genus $s$ and maximal Picard number. We show that these examples are elliptic surfaces, proving that elliptic surfaces of maximal Picard number and arbitrary geometric genus may be embedded as quasismooth hypersurfaces in weighted projective space.