Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of $\mathbb A ^2$
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912897325596672 |
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| author | Oszer, Piotr |
| author_facet | Oszer, Piotr |
| contents | Using Hilbert-Burch matrices, we give an explicit description of the Białynicki-Birula cells on the Hilbert scheme of points on $\mathbb A ^2$ with isolated fixed points. If the fixed point locus is positive dimensional we obtain an étale rational map to the cell. We prove Conjecture 4.2 from arXiv:2309.06871 which we realize as a special case of our construction. We also show examples when the construction provides a rational étale map to the Hilbert scheme which is not contained in any Białynicki-Birula cell. Finally, we give an explicit description of the formal deformations of any ideal in the Hilbert scheme of points on the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of $\mathbb A ^2$ Oszer, Piotr Algebraic Geometry Commutative Algebra 14C05 (Primary), 13D02 (Secondary) Using Hilbert-Burch matrices, we give an explicit description of the Białynicki-Birula cells on the Hilbert scheme of points on $\mathbb A ^2$ with isolated fixed points. If the fixed point locus is positive dimensional we obtain an étale rational map to the cell. We prove Conjecture 4.2 from arXiv:2309.06871 which we realize as a special case of our construction. We also show examples when the construction provides a rational étale map to the Hilbert scheme which is not contained in any Białynicki-Birula cell. Finally, we give an explicit description of the formal deformations of any ideal in the Hilbert scheme of points on the plane. |
| title | Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of $\mathbb A ^2$ |
| topic | Algebraic Geometry Commutative Algebra 14C05 (Primary), 13D02 (Secondary) |
| url | https://arxiv.org/abs/2407.07993 |