Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913222499500032 |
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| author | Bertone, Cristina Cioffi, Francesca Orth, Matthias Seiler, Werner M. |
| author_facet | Bertone, Cristina Cioffi, Francesca Orth, Matthias Seiler, Werner M. |
| contents | We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications.
The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations.
The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11770 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases Bertone, Cristina Cioffi, Francesca Orth, Matthias Seiler, Werner M. Commutative Algebra Algebraic Geometry 13P10, 14C05, 14M05, 13H10, 13F55 We introduce the notion of a relative marked basis over quasi-stable ideals, together with constructive methods and a functorial interpretation, developing computational methods for the study of Hilbert schemes over quotients of polynomial rings. Then we focus on two applications. The first has a theoretical flavour and produces an explicit open cover of the Hilbert scheme when the quotient ring is Cohen-Macaulay on quasi-stable ideals. Together with relative marked bases, we use suitable general changes of variables which preserve the structure of the quasi-stable ideal, against the expectations. The second application has a computational flavour. When the quotient rings are Macaulay-Lex on quasi-stable ideals, we investigate the lex-point of the Hilbert schemes and find examples of both smooth and singular lex-points. |
| title | Open Covers and Lex Points of Hilbert schemes over quotient rings via relative marked bases |
| topic | Commutative Algebra Algebraic Geometry 13P10, 14C05, 14M05, 13H10, 13F55 |
| url | https://arxiv.org/abs/2203.11770 |