http://mathoverflow.net/questions/79512/non-degenerate-pairing-on-neron-severi-group
arXiv:1210.3720 Computing Néron-Severi groups and cycle class groups fra arXiv Front: math.AG av Bjorn Poonen, Damiano Testa, Ronald van Luijk Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron-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.
nLab page on Neron-Severi group