On jet schemes of determinantal varieties
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915363969564672 |
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| author | Chen, Yifan Zuo, Huaiqing |
| author_facet | Chen, Yifan Zuo, Huaiqing |
| contents | Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the $k$-th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix as $\mathscr{L}^{m,n}_{r,k}$. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are 3 while $k$ is 1, we establish a correspondence between the defining ideals of $\mathscr{L}^{m,n}_{r,k}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of $\mathscr{L}^{m,n}_{r,k}$ accordingly. Moreover, for general $\mathscr{L}^{m,n}_{r,k}$, \cite{12} provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components.
Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_22898 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On jet schemes of determinantal varieties Chen, Yifan Zuo, Huaiqing Algebraic Geometry 14M12 14E18 05E40 Determinantal varieties are important objects of study in algebraic geometry. In this paper, we will investigate them using the jet scheme approach. We have found a new connection for the Hilbert series between a determinantal variety and its jet schemes. We denote the $k$-th order jet scheme of the determinantal variety defined by $r$-minors in an $m \times n$ matrix as $\mathscr{L}^{m,n}_{r,k}$. For the special case where $m$, $n$, and $r$ are equal, and $m$ and $r$ are 3 while $k$ is 1, we establish a correspondence between the defining ideals of $\mathscr{L}^{m,n}_{r,k}$ and abstract simplicial complexes, proving their shellability and obtaining the Hilbert series of $\mathscr{L}^{m,n}_{r,k}$ accordingly. Moreover, for general $\mathscr{L}^{m,n}_{r,k}$, \cite{12} provides its irreducible decomposition. We further provide a specific polynomial family defining its irreducible components. Keywords. Determinantal varieties, jet schemes, shellability, Hilbert series. |
| title | On jet schemes of determinantal varieties |
| topic | Algebraic Geometry 14M12 14E18 05E40 |
| url | https://arxiv.org/abs/2506.22898 |