Hodge loci associated with linear subspaces intersecting in codimension one
Remke Kloosterman
TL;DR
The paper investigates when the Noether-Lefschetz loci NL(Pi1,Pi2) and NL(Pi1+λPi2) coincide near a smooth hypersurface X in P^{2k+1} containing two k-planes intersecting in codimension one. It translates Hodge-type conditions into tangent-space computations via the Jacobian ring S/J and the associated ideal I(γ), and analyzes the two-class deformation [Y1]+λ[Y2] using left kernels and Gram-matrix methods (Theorem thmTsp). The results separate the cases k=1 and k≥2: for k=1 and d≥5, NL(Pi1+λPi2) is nonreduced for all λ≠0,1, while for k≥2 there exist X for which NL(Pi1+λPi2)=NL(Pi1,Pi2) for all but finitely many λ (and potentially {0,1} when d≥6), with generic X yielding equality near X for λ≠0,1. The paper also provides explicit constructions to demonstrate the method and discusses embedded components arising from special split configurations, thereby clarifying when Noether-Lefschetz phenomena persist or degenerate in higher codimension.
Abstract
Let $X\subset \mathbb{P}^{2k+1}$ be a smooth hypersurface containing two k-dimensional linear spaces $Π_1,Π_2$ intersecting in codimension one. In this paper we study the question whether the Hodge loci $NL([Π_1]+λ[Π_2])$ and $NL([Π_1],[Π_2])$ coincide. This turns out to be the case in a neighborhood of $X$ if $X$ is very general on $NL([Π_1],[Π_2])$, $k>1$ and $λ\neq 0,1$. However, there exists a hypersurface $X$ for which $NL([Π_1],[Π_2])$ is smooth at $X$, but $NL([Π_1]+λ[Π_2])$ is singular for all $λ\neq0,1$. We expect that this is due to an embedded component of $NL([Π_1]+λ[Π_2])$. The case $k=1$ was treated before by Dan, in that case $NL([Π_1]+λ[Π_2])$ is nonreduced.