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no. 6
The non-existence of $\mathbb{F}_q$-translation spreads of $H(q^2)$
Bartoli, Daniele1; Giannoni, Alessandro2; Giulietti, Massimo1; Marino, Giuseppe2
1 University of Perugia, Department of Mathematics and Informatics, via Vanvitelli 1, Perugia, 06123 (Italy)
2 University of Naples “Federico II", Dipartimento di Matematica e Applicazioni, Via Cintia, Monte S.Angelo, Naples, 80126 (Italy)
Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1603-1615

Using the connection between translation spreads of the generalized hexagon $H(q)$ and linear sets (see Cardinali et al. in Eur J Comb 23:367–376, 2002; Lunardon and Polverino in J Algebraic Comb 18:255–262, 2003), the non-existence of $\mathbb{F}_q$-translation spreads of $H(q^2)$ when $p>3$, $q=p^h$, and $q$ is large enough is proven. This answers to a question posed in [Marino and Polverino in J Algebraic Comb 42:725-744, 2015].

Received:
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Accepted:
Published online:
DOI: 10.5802/alco.456
Classification: 51E23, 51E12, 14H50
Keywords: generalized hexagon, spread, twisted cubic, linear sets

Bartoli, Daniele 1; Giannoni, Alessandro 2; Giulietti, Massimo 1; Marino, Giuseppe 2

1 University of Perugia, Department of Mathematics and Informatics, via Vanvitelli 1, Perugia, 06123 (Italy)
2 University of Naples “Federico II", Dipartimento di Matematica e Applicazioni, Via Cintia, Monte S.Angelo, Naples, 80126 (Italy)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Bartoli, Daniele; Giannoni, Alessandro; Giulietti, Massimo; Marino, Giuseppe. The non-existence of $\mathbb{F}_q$-translation spreads of $H(q^2)$. Algebraic Combinatorics, Volume 8 (2025) no. 6, pp. 1603-1615. doi: 10.5802/alco.456

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