Hodge loci associated with linear subspaces intersecting in codimension one

Let X⊂P2k+1$X\subset \mathbf {P}^{2k+1}$ be a smooth hypersurface containing two k$k$‐dimensional linear spaces Π1,Π2$\Pi _1,\Pi _2$, such that dimΠ1∩Π2=k−1$\dim \Pi _1\cap \Pi _2=k-1$. In this paper, we study the question whether the Hodge loci NL([Π1]+λ[Π2])$\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$ and NL([Π1],[Π2])$\operatorname{NL}([\Pi _1],[\Pi _2])$ coincide. This turns out to be the case in a neighborhood of X$X$ if X$X$ is very general on NL([Π1],[Π2])$\operatorname{NL}([\Pi _1],[\Pi _2])$, k>1$k>1$, and λ≠0,1$\lambda \ne 0,1$. However, there exists a hypersurface X$X$ for which NL([Π1],[Π2])$\operatorname{NL}([\Pi _1],[\Pi _2])$ is smooth at X$X$, but NL([Π1]+λ[Π2])$\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$ is singular for all λ≠0,1$\lambda \ne 0,1$. We expect that this is due to an embedded component of NL([Π1]+λ[Π2])$\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$. The case k=1$k=1$ was treated before by Dan, in that case NL([Π1]+λ[Π2])$\operatorname{NL}([\Pi _1]+\lambda [\Pi _2])$ is nonreduced.