Affine holomorphic bundles over $\mathbb{P}^1_\mathbb{C}$ and apolar ideals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916576841695232 |
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| author | Bouchareb, Naoufal |
| author_facet | Bouchareb, Naoufal |
| contents | We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2 affine bundles over $\mathbb{P}^1_\mathbb{C}$ whose linearisation, viewed as locally free sheaf, is isomorphic to $ {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_1)\oplus {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_2)$ where $n_1>n_2$. We show that this moduli space can be identified with the "topological cokernel" of a morphism of linear spaces over the projective space $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ of binary forms of degree $l:= -2-n_2$, in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ defined by the level sets of the fibre dimension map is determined explicitly by $d:= n_1-n_2$ and the cactus rank stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Affine holomorphic bundles over $\mathbb{P}^1_\mathbb{C}$ and apolar ideals Bouchareb, Naoufal Algebraic Geometry Complex Variables 32L05, 32L10, 32G13, 14D20 We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2 affine bundles over $\mathbb{P}^1_\mathbb{C}$ whose linearisation, viewed as locally free sheaf, is isomorphic to $ {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_1)\oplus {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_2)$ where $n_1>n_2$. We show that this moduli space can be identified with the "topological cokernel" of a morphism of linear spaces over the projective space $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ of binary forms of degree $l:= -2-n_2$, in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ defined by the level sets of the fibre dimension map is determined explicitly by $d:= n_1-n_2$ and the cactus rank stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$. |
| title | Affine holomorphic bundles over $\mathbb{P}^1_\mathbb{C}$ and apolar ideals |
| topic | Algebraic Geometry Complex Variables 32L05, 32L10, 32G13, 14D20 |
| url | https://arxiv.org/abs/2410.18706 |