Hodge loci associated with linear subspaces intersecting in codimension one
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916650466410496 |
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| author | Kloosterman, Remke |
| author_facet | Kloosterman, Remke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10775 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hodge loci associated with linear subspaces intersecting in codimension one Kloosterman, Remke Algebraic Geometry 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. |
| title | Hodge loci associated with linear subspaces intersecting in codimension one |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2401.10775 |