Computing N\'eron-Severi groups and cycle class groups

TL;DR

This paper presents an algorithm to compute the Néron-Severi group and the rank of cycle class groups of smooth projective varieties, assuming the Tate conjecture and the computability of étale cohomology.

Contribution

It introduces a method to compute these groups under certain conjectural assumptions, advancing computational algebraic geometry.

Findings

Algorithm computes Néron-Severi groups assuming Tate conjecture

Calculates ranks of cycle class groups for any codimension p

Depends on the computability of étale cohomology with finite coefficients

Abstract

Assuming the Tate conjecture and the computability of \'etale cohomology with finite coefficients, we give an algorithm that computes the N\'eron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.