A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917381034475520 |
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| author | Nikolov, Erik |
| author_facet | Nikolov, Erik |
| contents | For a smooth projective variety $X$ of dimension $d \geq 5$ over an algebraically closed field $k$ of characteristic zero, it is shown in this paper that the bounded derived category of the Hilbert scheme of three points $X^{[3]}$ admits a semi-orthogonal sequence of length $\binom{d-3}{2}$. Each subcategory in this sequence is equivalent to the derived category of $X$ and realized as the image of a Fourier-Mukai transform along a Grassmannian bundle $\mathbb{G}$ over $X$ parametrizing planar subschemes in $X^{[3]}$. The main ingredient in the proof is the computation of the normal bundle of $\mathbb{G}$ in $X^{[3]}$. An analogous result for generalized Kummer varieties is deduced at the end. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points Nikolov, Erik Algebraic Geometry For a smooth projective variety $X$ of dimension $d \geq 5$ over an algebraically closed field $k$ of characteristic zero, it is shown in this paper that the bounded derived category of the Hilbert scheme of three points $X^{[3]}$ admits a semi-orthogonal sequence of length $\binom{d-3}{2}$. Each subcategory in this sequence is equivalent to the derived category of $X$ and realized as the image of a Fourier-Mukai transform along a Grassmannian bundle $\mathbb{G}$ over $X$ parametrizing planar subschemes in $X^{[3]}$. The main ingredient in the proof is the computation of the normal bundle of $\mathbb{G}$ in $X^{[3]}$. An analogous result for generalized Kummer varieties is deduced at the end. |
| title | A Semi-orthogonal Sequence in the Derived Category of the Hilbert Scheme of Three Points |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2404.12851 |