Explicit deformation of a spider algebra to a curvilinear scheme via Möbius generators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910036777762816 |
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| author | Turturean, David |
| author_facet | Turturean, David |
| contents | We construct an explicit flat one-parameter family of 22-dimensional Artinian $k$-algebras whose special fibre is the spider algebra $k[x,y,z]/(x^8, y^8, z^8, xy, xz, yz)$ and whose generic fibre is the curvilinear algebra $k[t]/(t^{22})$. The construction uses Möbius generators $u_a = t/(1-at)$ inside the curvilinear ring together with a divided-difference change of coordinates, and produces the family via a weighted Rees degeneration with integer coefficients. This gives an explicit one-parameter family witnessing, for this spider ideal, the general phenomenon proved by Bérczi-Svendsen that every monomial subscheme of $\mathbb{C}^d$ lies in the curvilinear component of the Hilbert scheme of points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00886 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit deformation of a spider algebra to a curvilinear scheme via Möbius generators Turturean, David Algebraic Geometry Commutative Algebra 14C05 (Primary), 13D10, 14B07, 13H10, 13E10 (Secondary) We construct an explicit flat one-parameter family of 22-dimensional Artinian $k$-algebras whose special fibre is the spider algebra $k[x,y,z]/(x^8, y^8, z^8, xy, xz, yz)$ and whose generic fibre is the curvilinear algebra $k[t]/(t^{22})$. The construction uses Möbius generators $u_a = t/(1-at)$ inside the curvilinear ring together with a divided-difference change of coordinates, and produces the family via a weighted Rees degeneration with integer coefficients. This gives an explicit one-parameter family witnessing, for this spider ideal, the general phenomenon proved by Bérczi-Svendsen that every monomial subscheme of $\mathbb{C}^d$ lies in the curvilinear component of the Hilbert scheme of points. |
| title | Explicit deformation of a spider algebra to a curvilinear scheme via Möbius generators |
| topic | Algebraic Geometry Commutative Algebra 14C05 (Primary), 13D10, 14B07, 13H10, 13E10 (Secondary) |
| url | https://arxiv.org/abs/2603.00886 |