Weighted surfaces with maximal Picard number

TL;DR

This paper extends Shioda's algorithm to weighted Delsarte surfaces in weighted projective space, providing criteria for maximal Picard number and constructing examples with arbitrary geometric genus, revealing their elliptic nature.

Contribution

It introduces a criterion linking maximal Picard number to automorphism groups for weighted Delsarte surfaces and constructs explicit examples with arbitrary genus.

Findings

Weighted Delsarte surfaces can have maximal Picard number under specific conditions.

Examples of elliptic surfaces with arbitrary geometric genus and maximal Picard number are constructed.

Such surfaces can be embedded as quasismooth hypersurfaces in weighted projective space.

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.