Computing Néron–Severi groups and cycle class groups

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 $p$.

Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support

Let νd:ℙm→ℙn, n:=(n+dn)-1, denote the degree d Veronese embedding of ℙm. For any P∈ℙn, let sr(P) be the minimal cardinality of S⊂νd(ℙm) such that P∈〈S〉. Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯(S) powers Ld with L a linear form (or as a sum of ♯(S) d-powers of vectors). Here we fix an integral variety T⊊ℙm and P∈〈νd(T)〉 and study a similar decomposition with S⊈T and ♯(S) minimal. For instance, if T is a linear subspace, then we prove that ♯(S)≥♯(S∩T)+d+1 and classify all (S,P) such that ♯(S)-♯(S∩T)≤2d-1.