2006/05/12 by Vincenzo Di Gennaro, Di Gennaro, Vincenzo, Davide Franco +1
Computer Science · Mathematics · #14B05 #14C30 #14D05 #14J17 #14M10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14B05 #msc:14C30 #msc:14D05 #msc:14J17 #msc:14M10
paper · pdf · doi:10.48550/arxiv.math/0605341
24 pages
arxiv created 2006/05/12 · openalex publication_date 2006/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a projective variety Z and for any integer p, define the p-th Néron-Severi group NSp(Z) of Z as the image of the cycle map Ap(Z)→ H2p(Z; ℂ). Now let X⊂ \Ps2m+1 (m≥ 1) be a projective variety of dimension 2m-1, with isolated singularities, complete intersection of a smooth hypersurface of degree k, with a hypersurface of degree n>max\k, 2m+1\, and let F be a general hypersurface of degree n containing X. We prove that the natural map NSm(X)→ NSm(F) is surjective, and that if dim NSm(F)=1 then dim NSm(X)=1. In particular dim NSm(X)=1 if and only if dim NSm(F)=1. When X is a threefold (i.e. m=2) we deduce a new characterization for the factoriality of X, i.e. that X is factorial if and only if dim NS2(F)=1. This allows us to give examples of factorial threefolds, in some case with many singularities. During the proof of the announced results, we show that the quotient of the middle cohomology of F by the cycle classes coming from X is irreducible under the monodromy action induced by the hypersurfaces of degree n containing X. As consequences we deduce a Noether-Lefschetz Theorem for a projective complete intersection with isolated singularities, and, also using a recent result on codimension two Hodge conjecture, in the case X⊂ \Ps5 is a threefold as before, we deduce that the general hypersurface F of degree n containing X verifies Hodge conjecture.