Faithfully flat quotient morphisms by $G_a$-actions on factorial affine varieties
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908697140133888 |
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| author | Masuda, Kayo |
| author_facet | Masuda, Kayo |
| contents | Let $X$ be a factorial complex affine variety of dimension $\ge 3$ with an algebraic action of the additive group $G_a$. Let $π: X \to Y$ be the algebraic quotient morphism where we assume $Y$ is an affine variety. When $π$ is faithfully flat, we investigate $π$ by $G_a$-equivariant affine modifications and give criteria for $π$ to be a trivial $\mathbb A^1$-bundle. For a smooth acyclic fourfold $X$ with a free $G_a$-action and a $G_a$-equivariant $\mathbb A^3$-fibration $f : X \to \mathbb A^1$ where $G_a$ acts trivially on $\mathbb A^1$, we give a criterion for the algebraic quotient $Y$ to be isomorphic to $\mathbb A^3$ with $f$ as a coordinate. Together with a criterion for $π: X \to Y$ to be a trivial $\mathbb A^1$-bundle, we obtain a sufficient condition for $X\cong Y\times \mathbb A^1\cong \mathbb A^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06687 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Faithfully flat quotient morphisms by $G_a$-actions on factorial affine varieties Masuda, Kayo Algebraic Geometry 14R20 Let $X$ be a factorial complex affine variety of dimension $\ge 3$ with an algebraic action of the additive group $G_a$. Let $π: X \to Y$ be the algebraic quotient morphism where we assume $Y$ is an affine variety. When $π$ is faithfully flat, we investigate $π$ by $G_a$-equivariant affine modifications and give criteria for $π$ to be a trivial $\mathbb A^1$-bundle. For a smooth acyclic fourfold $X$ with a free $G_a$-action and a $G_a$-equivariant $\mathbb A^3$-fibration $f : X \to \mathbb A^1$ where $G_a$ acts trivially on $\mathbb A^1$, we give a criterion for the algebraic quotient $Y$ to be isomorphic to $\mathbb A^3$ with $f$ as a coordinate. Together with a criterion for $π: X \to Y$ to be a trivial $\mathbb A^1$-bundle, we obtain a sufficient condition for $X\cong Y\times \mathbb A^1\cong \mathbb A^4$. |
| title | Faithfully flat quotient morphisms by $G_a$-actions on factorial affine varieties |
| topic | Algebraic Geometry 14R20 |
| url | https://arxiv.org/abs/2512.06687 |