Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918002494013440 |
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| author | Desjardins, Julie Winter, Rosa |
| author_facet | Desjardins, Julie Winter, Rosa |
| contents | The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $S$. A natural question arises when studying the configuration of these curves: if a point on $S$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11659 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one Desjardins, Julie Winter, Rosa Algebraic Geometry Number Theory The blow-up of the anticanonical base point on a del Pezzo surface $S$ of degree 1 gives rise to a rational elliptic surface $\mathscr{E}$ with only irreducible fibers. The sections of minimal height of $\mathscr{E}$ are in correspondence with the $240$ exceptional curves on $S$. A natural question arises when studying the configuration of these curves: if a point on $S$ is contained in 'many' exceptional curves, it is torsion on its fiber on $\mathscr{E}$? In 2005, Kuwata proved for the analogous question on del Pezzo surfaces of degree $2$, where there are 56 exceptional curves, that if 'many' equals $4$ or more, the answer is yes. In this paper, we prove that for del Pezzo surfaces of degree 1, the answer is yes if 'many' equals $9$ or more. Moreover, we give counterexamples where a \textsl{non}-torsion point lies in the intersection of $7$ exceptional curves. We give partial results for the still open case of 8 intersecting exceptional curves. |
| title | Torsion points and concurrent exceptional curves on del Pezzo surfaces of degree one |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2210.11659 |