Hodge loci associated with linear subspaces intersecting in codimension one

Abstract

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