On the Tate conjecture for the Fano surfaces of cubic threefolds

TL;DR

This paper proves that Fano surfaces of smooth cubic threefolds satisfy the Tate conjecture over fields of finite type over the prime field, extending understanding of algebraic cycles in this geometric context.

Contribution

It establishes the Tate conjecture for Fano surfaces of cubic threefolds in new settings, specifically over fields of finite type over the prime field and characteristic not 2.

Findings

Fano surfaces satisfy the Tate conjecture in the specified setting

The proof applies to fields of finite type over the prime field

Characteristic not 2 is a necessary condition

Abstract

A Fano surface of a smooth cubic threefold X in P^4 parametrizes the lines on X. In this note, we prove that a Fano surface satisfies the Tate conjecture over a field of finite type over the prime field and characteristic not 2.