Rational lines on smooth cubic surfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866915177748758528 |
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| author | McKean, Stephen |
| author_facet | McKean, Stephen |
| contents | We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_08217 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Rational lines on smooth cubic surfaces McKean, Stephen Algebraic Geometry 14N15, 14G25 We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field. |
| title | Rational lines on smooth cubic surfaces |
| topic | Algebraic Geometry 14N15, 14G25 |
| url | https://arxiv.org/abs/2101.08217 |