Hodge loci associated with linear subspaces intersecting in codimension one

Let X in P^{2k+1} be a smooth hypersurface containing two k-dimensional linear spaces P_1 , P_2, such that dim P_1 \cap P_2 =k-1. In this paper, we study the question whether the Hodge loci NL([P_1 ] + lambda[P_2 ]) and NL([P_2], [P_2]) coincide. This turns out to be the case in a neighborhood of X if X is very general on NL([P_1 ], [P_2 ]), k>1, and lambda <> 0, 1. However, there exists a hypersurface X for which NL([P_1 ], [P_2 ]) is smooth at X, but NL([P_1 ] + lambda[P_2 ]) is singular for all lambda<> 0, 1. We expect that this is due to an embedded component of NL([P_1 ] + lambda[P_2 ]). The case k=1 was treated before by Dan, in that case NL([P_1 ] + lambda [P_2 ]) is nonreduced