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| Format: | Preprint |
| Published: |
2021
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| Online Access: | https://arxiv.org/abs/2105.00365 |
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| _version_ | 1866916980540309504 |
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| author | Kiermaier, Michael |
| author_facet | Kiermaier, Michael |
| contents | Arguably, the most important open problem in the theory of $q$-analogs of designs is the question for the existence of a $q$-analog $D$ of the Fano plane. It is undecided for every single prime power value $q \geq 2$.
A point $P$ is called an $α$-point of $D$ if the derived design of $D$ in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must always exist at least one non-$α$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$α$-points must form a blocking set with respect to the hyperplanes.
In this article, we show that a hyperplane consisting only of $α$-points implies the existence of a partiton of the symplectic generalized quadrangle $W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes $q$ and all even values of $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_00365 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On $α$-points of $q$-analogs of the Fano plane Kiermaier, Michael Combinatorics Arguably, the most important open problem in the theory of $q$-analogs of designs is the question for the existence of a $q$-analog $D$ of the Fano plane. It is undecided for every single prime power value $q \geq 2$. A point $P$ is called an $α$-point of $D$ if the derived design of $D$ in $P$ is a geometric spread. In 1996, Simon Thomas has shown that there must always exist at least one non-$α$-point. For the binary case $q = 2$, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-$α$-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $α$-points implies the existence of a partiton of the symplectic generalized quadrangle $W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes $q$ and all even values of $q$. |
| title | On $α$-points of $q$-analogs of the Fano plane |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2105.00365 |