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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2601.05715 |
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| _version_ | 1866908754991120384 |
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| author | Kaygun, Atabey |
| author_facet | Kaygun, Atabey |
| contents | We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (à la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map $κ^{inc}_{2,μ}\colon H^2_{inc}(μ)\to H^3_{inc}(μ)$ that controls second-order lifts modulo isotriviality. If $μ$ is smooth point in a reduced component and $(κ^{inc}_{2,μ})^{-1}(0)=\{0\}$, then the $G$-orbit of $μ$ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05715 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Rigidity in Moduli Stacks of Algebras Kaygun, Atabey Algebraic Geometry 14D23 13D10 We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (à la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map $κ^{inc}_{2,μ}\colon H^2_{inc}(μ)\to H^3_{inc}(μ)$ that controls second-order lifts modulo isotriviality. If $μ$ is smooth point in a reduced component and $(κ^{inc}_{2,μ})^{-1}(0)=\{0\}$, then the $G$-orbit of $μ$ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish. |
| title | Geometric Rigidity in Moduli Stacks of Algebras |
| topic | Algebraic Geometry 14D23 13D10 |
| url | https://arxiv.org/abs/2601.05715 |