Sylvester--Gallai configurations on algebraic curves in C^2
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916924966830080 |
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| author | Cohen, Alex |
| author_facet | Cohen, Alex |
| contents | The Sylvester-Gallai theorem says that for any finite set of non-collinear points in $\R^2$, there is some line passing through exactly two points of the set. Over the complex numbers, this theorem fails: there are finite configurations with the property that any line through two points also passes through a third. Only one infinite class of examples (the Fermat configurations) is known, and it is a folklore conjecture that this is the only infinite class of examples. We prove this conjecture in the ``99\% structure'' case where we assume most of the points lie on a low degree algebraic curve. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21241 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sylvester--Gallai configurations on algebraic curves in C^2 Cohen, Alex Combinatorics The Sylvester-Gallai theorem says that for any finite set of non-collinear points in $\R^2$, there is some line passing through exactly two points of the set. Over the complex numbers, this theorem fails: there are finite configurations with the property that any line through two points also passes through a third. Only one infinite class of examples (the Fermat configurations) is known, and it is a folklore conjecture that this is the only infinite class of examples. We prove this conjecture in the ``99\% structure'' case where we assume most of the points lie on a low degree algebraic curve. |
| title | Sylvester--Gallai configurations on algebraic curves in C^2 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.21241 |