Hodge loci associated with linear subspaces intersecting in codimension one

Abstract Let be a smooth hypersurface containing two ‐dimensional linear spaces , such that . In this paper, we study the question whether the Hodge loci and coincide. This turns out to be the case in a neighborhood of if is very general on , , and . However, there exists a hypersurface for which is smooth at , but is singular for all . We expect that this is due to an embedded component of . The case was treated before by Dan, in that case is nonreduced.