Quintic surfaces with maximum and other Picard numbers

TL;DR

This paper constructs specific complex quintic surfaces with maximal and other high Picard numbers, advancing understanding of their geometric and arithmetic properties through explicit examples and zeta function analysis.

Contribution

It provides the first example of a complex quintic surface with maximum Picard number and introduces techniques to produce surfaces with various high Picard numbers.

Findings

First example of a quintic surface with Picard number 45

Determined the zeta function of the maximal Picard number surface

Produced new examples of quintic surfaces with previously unachieved Picard numbers

Abstract

This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in IP^3 with maximum Picard number 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.