Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866912021355692032 |
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| author | Dallaporta, William |
| author_facet | Dallaporta, William |
| contents | Let $S$ be a scheme such that $2$ is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over $S$, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over $S$ and isomorphism classes of pairs $(\mathcal{C},\mathcal{M})$ where $\mathcal{C}$ is a quadratic algebra over $S$ and $\mathcal{M}$ is an invertible $\mathcal{C}$-module. Unexpectedly, examples suggest that a refinement of Wood's bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over $S$; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_13422 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms Dallaporta, William Number Theory Algebraic Geometry 11E16, 11R29, 14C22 (Primary) Let $S$ be a scheme such that $2$ is not a zero divisor. In this paper, we address the following question: given a quadratic algebra over $S$, how can we parametrize its Picard group in terms of quadratic forms? In 2011, Wood established a set-theoretical bijection between isomorphism classes of primary binary quadratic forms over $S$ and isomorphism classes of pairs $(\mathcal{C},\mathcal{M})$ where $\mathcal{C}$ is a quadratic algebra over $S$ and $\mathcal{M}$ is an invertible $\mathcal{C}$-module. Unexpectedly, examples suggest that a refinement of Wood's bijection is needed in order to parametrize Picard groups. This is why we start by classifying quadratic algebras over $S$; this is achieved by using two invariants, the discriminant and the parity. Extending the notion of orientation of quadratic algebras to the non-free case is another key step, eventually leading us to the desired parametrization. All along the paper, we illustrate various notions and obstructions with a wide range of examples. |
| title | Recovering the Picard group of quadratic algebras from Wood's binary quadratic forms |
| topic | Number Theory Algebraic Geometry 11E16, 11R29, 14C22 (Primary) |
| url | https://arxiv.org/abs/2111.13422 |