Generically log smooth families via generators and relations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917282933899264 |
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| author | Felten, Simon |
| author_facet | Felten, Simon |
| contents | Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\barρ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generically log smooth families via generators and relations Felten, Simon Algebraic Geometry 14-04, 14A21, 14B05, 14D06, 14Q15 Let $f\colon X \to \mathbb{A}^1_t$ be an affine flat morphism of finite type, and let $V = f^{-1}(0)$. Then, we obtain a morphism of log schemes $f\colon (X|V) \to (\mathbb{A}^1_t|0)$. In this article, we develop algorithmic tools to study the log-geometric properties of $f$ by means of a presentation \[Γ(X,\mathcal{O}_X) = \Bbbk[t,x_1,\ldots,x_n]/(f_1,\ldots,f_r).\] We obtain similar tools for projective flat morphisms when the homogeneous coordinate ring is given by generators and relations. We provide an implementation of our algorithms in Macaulay2. In a slightly different direction, we give some results on the sheaf $\mathcal{LS}_V$ of log smooth structures on a toroidal crossing scheme $(V,\mathcal{P},\barρ)$. |
| title | Generically log smooth families via generators and relations |
| topic | Algebraic Geometry 14-04, 14A21, 14B05, 14D06, 14Q15 |
| url | https://arxiv.org/abs/2602.17617 |