On the proportion of transverse-free plane curves
| Autor: | Asgarli, Shamil, Freidin, Brian |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | We study the asymptotic proportion of smooth plane curves over a finite field $\mathbb{F}_q$ which are tangent to every line defined over $\mathbb{F}_q$. This partially answers a question raised by Charles Favre. Our techniques include applications of Poonen's Bertini theorem and Schrijver's theorem on perfect matchings in regular bipartite graphs. Our main theorem implies that a random smooth plane curve over $\mathbb{F}_q$ admits a transverse $\mathbb{F}_q$-line with very high probability. Comment: 19 pages |
| Databáze: | arXiv |
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