Monomial algebras and $\mathbb{G}_a^n$-equivariant embeddings into toric varieties
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915969282080768 |
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| author | Chernov, Alexander |
| author_facet | Chernov, Alexander |
| contents | An induced additive action on a projective variety $X\subseteq\mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^n$ on $X$ with an open orbit that can be extended to a regular action on $\mathbb{P}^n$. Such actions are known to correspond to pairs $(A, U)$, where $A$ is a local algebra and $U$ is a generating subspace lying in the maximal ideal. This paper studies additive actions on projective toric varieties, with a particular focus on toric surfaces. We prove that for any linearly normal toric variety equipped with a torus-normalized additive action, the associated pair consists of a monomial algebra and a subspace spanned by variables. Also we describe pairs that correspond to additive actions on toric surfaces in low-dimensional projective spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_22820 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monomial algebras and $\mathbb{G}_a^n$-equivariant embeddings into toric varieties Chernov, Alexander Algebraic Geometry Primary 14M25, 14L30, Secondary 13F55, 52B20 An induced additive action on a projective variety $X\subseteq\mathbb{P}^n$ is a regular action of the group $\mathbb{G}_a^n$ on $X$ with an open orbit that can be extended to a regular action on $\mathbb{P}^n$. Such actions are known to correspond to pairs $(A, U)$, where $A$ is a local algebra and $U$ is a generating subspace lying in the maximal ideal. This paper studies additive actions on projective toric varieties, with a particular focus on toric surfaces. We prove that for any linearly normal toric variety equipped with a torus-normalized additive action, the associated pair consists of a monomial algebra and a subspace spanned by variables. Also we describe pairs that correspond to additive actions on toric surfaces in low-dimensional projective spaces. |
| title | Monomial algebras and $\mathbb{G}_a^n$-equivariant embeddings into toric varieties |
| topic | Algebraic Geometry Primary 14M25, 14L30, Secondary 13F55, 52B20 |
| url | https://arxiv.org/abs/2510.22820 |