G-torsors and universal torsors over nonsplit del Pezzo surfaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929392454729728 |
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| author | Derenthal, Ulrich Hoffmann, Norbert |
| author_facet | Derenthal, Ulrich Hoffmann, Norbert |
| contents | Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Néron-Severi torus. Let $\mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $\mathcal{T}$ over $S_L$. We prove that $\mathcal{G}$ does not descend to S unless $\mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $\mathcal{G}$ always descend to singular del Pezzo surfaces over $\mathbb{C}$ from their desingularizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_08137 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | G-torsors and universal torsors over nonsplit del Pezzo surfaces Derenthal, Ulrich Hoffmann, Norbert Algebraic Geometry Number Theory 14J26 (Primary), 14L30, 11E57, 14G27 (Secondary) Let S be a smooth del Pezzo surface that is defined over a field K and splits over a Galois extension L. Let G be either the split reductive group given by the root system of $S_L$ in Pic $S_L$, or a form of it containing the Néron-Severi torus. Let $\mathcal{G}$ be the G-torsor over $S_L$ obtained by extension of structure group from a universal torsor $\mathcal{T}$ over $S_L$. We prove that $\mathcal{G}$ does not descend to S unless $\mathcal{T}$ does. This is in contrast to a result of Friedman and Morgan that such $\mathcal{G}$ always descend to singular del Pezzo surfaces over $\mathbb{C}$ from their desingularizations. |
| title | G-torsors and universal torsors over nonsplit del Pezzo surfaces |
| topic | Algebraic Geometry Number Theory 14J26 (Primary), 14L30, 11E57, 14G27 (Secondary) |
| url | https://arxiv.org/abs/2109.08137 |