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arXiv · 2401.10775

Hodge loci associated with linear subspaces intersecting in codimension one

Remke Kloosterman
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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.

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Remke Kloosterman. 2024-01-19. Hodge loci associated with linear subspaces intersecting in codimension one. https://doi.org/10.1002/mana.202400066

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