On line-parallelisms of PG(3, q)
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| Format: | Preprint |
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2025
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| _version_ | 1866909653814738944 |
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| author | Pavese, Francesco Santonastaso, Paolo |
| author_facet | Pavese, Francesco Santonastaso, Paolo |
| contents | Let $\mathrm{PG}(3, q)$ denote the three-dimensional projective space over the finite field with $q$ elements. A line-spread of $\mathrm{PG}(3, q)$ is a collection $\mathcal{S}$ of mutually skew lines such that every point of $\mathrm{PG}(3, q)$ lies on exactly one line of $\mathcal{S}$. A parallelism of $\mathrm{PG}(3, q)$ is a set $Π$ of mutually skew line-spreads of $\mathrm{PG}(3, q)$ such that every line of $\mathrm{PG}(3, q)$ is contained in precisely one line-spread of $Π$. For a Desarguesian spread $\mathcal{D}$ and an elementary abelian group $E$ of order $q^2$ that stabilizes $\mathcal{D}$ and one of its lines, let $\mathcal{T}$ be the class of parallelisms of $\mathrm{PG}(3, q)$ admitting $E$, and comprising $\mathcal{D}$ and $q^2+q$ Hall spreads, each of which is obtained by switching one of the $q^2+q$ reguli of $\mathcal{D}$ through its $E$-fixed line. In this paper, the parallelisms in $\mathcal{T}$ are characterized geometrically and enumerated. Moreover, it is shown that $\mathcal{T}$ contains at least $Θ(q^{q-1} q!)$ mutually inequivalent parallelisms for $q$ even, and at least $Θ(q^{2q-3})$ mutually inequivalent parallelisms when $q$ is odd. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_16271 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On line-parallelisms of PG(3, q) Pavese, Francesco Santonastaso, Paolo Combinatorics 51E23, 05B40, 05B25 Let $\mathrm{PG}(3, q)$ denote the three-dimensional projective space over the finite field with $q$ elements. A line-spread of $\mathrm{PG}(3, q)$ is a collection $\mathcal{S}$ of mutually skew lines such that every point of $\mathrm{PG}(3, q)$ lies on exactly one line of $\mathcal{S}$. A parallelism of $\mathrm{PG}(3, q)$ is a set $Π$ of mutually skew line-spreads of $\mathrm{PG}(3, q)$ such that every line of $\mathrm{PG}(3, q)$ is contained in precisely one line-spread of $Π$. For a Desarguesian spread $\mathcal{D}$ and an elementary abelian group $E$ of order $q^2$ that stabilizes $\mathcal{D}$ and one of its lines, let $\mathcal{T}$ be the class of parallelisms of $\mathrm{PG}(3, q)$ admitting $E$, and comprising $\mathcal{D}$ and $q^2+q$ Hall spreads, each of which is obtained by switching one of the $q^2+q$ reguli of $\mathcal{D}$ through its $E$-fixed line. In this paper, the parallelisms in $\mathcal{T}$ are characterized geometrically and enumerated. Moreover, it is shown that $\mathcal{T}$ contains at least $Θ(q^{q-1} q!)$ mutually inequivalent parallelisms for $q$ even, and at least $Θ(q^{2q-3})$ mutually inequivalent parallelisms when $q$ is odd. |
| title | On line-parallelisms of PG(3, q) |
| topic | Combinatorics 51E23, 05B40, 05B25 |
| url | https://arxiv.org/abs/2506.16271 |