Computing Picard Schemes
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918301603463168 |
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| author | Kweon, Hyuk Jun Venkatesh, Madhavan |
| author_facet | Kweon, Hyuk Jun Venkatesh, Madhavan |
| contents | We present an algorithm to compute the torsion component $\mathrm{Pic}^τX$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^τX$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^τX$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $π^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16505 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Computing Picard Schemes Kweon, Hyuk Jun Venkatesh, Madhavan Algebraic Geometry Number Theory 14C22, 14F20, 14Q15 We present an algorithm to compute the torsion component $\mathrm{Pic}^τX$ of the Picard scheme of a smooth projective variety $X$ over a field $k$. Specifically, we describe $\mathrm{Pic}^τX$ as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on $\mathrm{Pic}^τX$. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group $π^{\mathrm{{e}t}}_1(X_{\bar{k}}, x)^{\mathrm{ab}}$. Moreover, we determine the Galois module structure of the first étale cohomology groups $H^1_{\mathrm{{e}t}}(X_{\bar{k}}, \mathbb{Z}/n\mathbb{Z})$ without requiring $n$ to be prime to the characteristic of $k$. |
| title | Computing Picard Schemes |
| topic | Algebraic Geometry Number Theory 14C22, 14F20, 14Q15 |
| url | https://arxiv.org/abs/2601.16505 |