Hodge loci associated with linear subspaces intersecting in codimension one

Abstract

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

arXiv admin note: text overlap with arXiv:2312.12363

References in corpus (1)

• Periods of Complete Intersection Algebraic Cycles