Laurent polynomials and deformations of non-isolated Gorenstein toric sigularities
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915493178245120 |
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| author | Filip, Matej |
| author_facet | Filip, Matej |
| contents | We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P, \partial X_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $\newt(f) = P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{\newt(f)}, \partial X_{\newt(f)})$ over $\CC[[\bfTT_f]]$, where $\bfTT_f$ is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if $f$ is $0$-mutable. The Kodaira--Spencer map of the constructed deformation is injective, and if $f$ is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_04486 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Laurent polynomials and deformations of non-isolated Gorenstein toric sigularities Filip, Matej Algebraic Geometry We establish a correspondence between one-parameter deformations of an affine Gorenstein toric pair $(X_P, \partial X_P)$, defined by a polytope $P$, and mutations of a Laurent polynomial $f$ with Newton polytope $\newt(f) = P$. For a Laurent polynomial $f$ in two variables, we construct a formal deformation of the three-dimensional Gorenstein toric pair $(X_{\newt(f)}, \partial X_{\newt(f)})$ over $\CC[[\bfTT_f]]$, where $\bfTT_f$ is the set of deformation parameters arising from mutations. The general fibre of this deformation is smooth if and only if $f$ is $0$-mutable. The Kodaira--Spencer map of the constructed deformation is injective, and if $f$ is maximally mutable, then the deformation cannot be nontrivially extended to a larger smooth base space. |
| title | Laurent polynomials and deformations of non-isolated Gorenstein toric sigularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2504.04486 |