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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.13084 |
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Table of Contents:
- We establish a group-action version of the Szemerédi-Trotter theorem over any field, extending Bourgain's result for the group $\mathrm{SL}_2(k)$. As an Elekes-Szabó-type application, we obtain quantitative bounds on the number of collinear triples on reducible cubic surfaces in $\mathbb{P}^3(k)$, where $k = \mathbb{F}_{q}$ and $k = \mathbb{C}$, thereby improving a recent result by Bays, Dobrowolski, and the second author.