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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2012
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1209.2026 |
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| _version_ | 1866929464107073536 |
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| author | Evain, Laurent Lederer, Mathias |
| author_facet | Evain, Laurent Lederer, Mathias |
| contents | The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are smooth in dimension $d=2$. We prove that there is a schematic structure in higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense that it represents a functor. Let $ρ_i:H^n(\mathbb{A}^d)\rightarrow {\rm Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$ coordinate. We prove that a Bialynicki-Birula scheme associated with an action of a torus $T$ is schematically included in the fiber $ρ_i^{-1}(0)$ if the ${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functors parametrizing families of ideals with a prescribed initial ideal are representable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1209_2026 |
| institution | arXiv |
| publishDate | 2012 |
| record_format | arxiv |
| spellingShingle | Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors Evain, Laurent Lederer, Mathias Algebraic Geometry 14C05 The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are smooth in dimension $d=2$. We prove that there is a schematic structure in higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense that it represents a functor. Let $ρ_i:H^n(\mathbb{A}^d)\rightarrow {\rm Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$ coordinate. We prove that a Bialynicki-Birula scheme associated with an action of a torus $T$ is schematically included in the fiber $ρ_i^{-1}(0)$ if the ${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functors parametrizing families of ideals with a prescribed initial ideal are representable. |
| title | Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors |
| topic | Algebraic Geometry 14C05 |
| url | https://arxiv.org/abs/1209.2026 |