The Barth quintic surface has Picard number 41

TL;DR

This paper proves that Barth's quintic surface has a Picard number of 41, computes its Neron-Severi group, and verifies the Tate conjecture through reductions in positive characteristic.

Contribution

It establishes the Picard number of Barth's quintic surface as 41 and computes its Neron-Severi group, advancing understanding of its geometric properties.

Findings

Picard number of the surface is 41

Neron-Severi group computed up to 2-power index

Tate conjecture verified for reductions

Abstract

This paper investigates a specific smooth quintic surface suggested by Barth for it contains the current record of 75 lines over the complex numbers. Our main incentive is to prove that the complex quintic has Picard number 41, and to compute the Neron-Severi group up to a 2-power index. We also compute Picard numbers for reductions to positive characteristic and verify the Tate conjecture.