Hodge loci associated with linear subspaces intersecting in codimension one

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