Picard numbers of quintic surfaces

TL;DR

This paper determines all possible Picard numbers for complex quintic surfaces over the rationals, showing every number from 1 to 45 can occur, using arithmetic deformations, K3 surfaces, and automorphisms.

Contribution

It completely solves the Picard number problem for quintic surfaces, establishing the realizability of all numbers from 1 to 45 over the rationals.

Findings

Every integer from 1 to 45 is realizable as a Picard number of a quintic surface.

The main technique involves arithmetic deformations of Delsarte surfaces.

Additional methods include the use of K3 surfaces and wild automorphisms.

Abstract

We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over the rationals. Our main technique consists in arithmetic deformations of Delsarte surfaces, but we also use K3 surfaces and wild automorphisms.