\(\alpha\)-flokki and partial \(\alpha\)-flokki
In this paper, the authors generalize the relationship between flocks of quadratic cone and spreads of $\mathrm{PG}(3,q)$ by remarking that a regulus $\mathcal{R}$ of $\mathrm{PG}(3,q)$ is a particular derivable partial spread which is fixed by an elation group $E$ of order $q$, whose axis $X$ belongs to the regulus and $\mathcal{R}\setminus \{X\}$ is an orbit of $E$. \par Considering homogeneous coordinates $(x_0, x_1, x_2, x_3)$ of $\mathrm{PG}(3,q)$ and an automorphism $\alpha$ of $\mathbb{F}_q$ we define the $\alpha$-cone $C_{\alpha}$ as $x_0^{\alpha} x_1 = x_2^{\alpha +1}$ with vertex $(0, 0, 0, 1)$. A set of planes of $\mathrm{PG}(3,q)$ which partition the non-vertex points of $C_{\alpha}$ will be called an $\alpha$-flokki. The intersections are called $\alpha$-conics. \par We note that for $\alpha=1$, $\mathcal{C}_{\alpha}$ is a quadratic cone and 1-flokki are flocks of a quadratic cone. \par Suppose that $\mathbb{C}=\{\begin{pmatrix}u +g(t) &f(t)\\ t& u^{\alpha}\end{pmatrix}\mid u,t \in \mathbb{F}_q\}$ is a spread set such that $f(0)=g(0)=0$. If $F(\infty)=\{(0,0,y_1,y_2)\mid y_1,y_1 \in \mathbb{F}_q\}$ and \[ F(u,t)=\{(x_1,x_2,y_1,y_2)\in \mathbb{F}_q^4 \mid (y_1,y_2)=(x_1,x_2)\begin{pmatrix} u &f(t)\\ t& u^{\alpha}+ g(t)\end{pmatrix}\}, \] let $\mathcal{S}=\{F(u,t)\mid u,t \in \mathbb{F}_q\}\cup \{F(\infty)\}$ be the associated spread of $\mathrm{PG}(3,q)$. Such a spread is fixed by the elation group $E$ of order $q$ defined by the maps \[ (x_1,x_2,y_1,y_2)\mapsto (x_1,x_2,y_1+x_1u,y_2+x_2u^{\alpha}) \] with axis $F(\infty)$ and the partial spreads $\mathcal{D}_t=\{F(u,t)\mid u \in \mathbb{F}_q\}\cup \{F(\infty)\}$ are derivable partial spreads such that $\mathcal{D}_t\setminus \{F(\infty)\}$ is an orbit of $E$. \par Then, the planes $\pi_t : x_0t-x_1f(t)^{\alpha}+ x_2g(t)^{\alpha} + x_3 = 0$ ($t\in \mathbb{F}_q$) define an $\alpha$-flokki. \par The authors prove that there is a bijection between $\alpha$-flokki and spreads of $\mathrm{PG}(3,q)$ which have an elation group $E$ of order $q$ one of whose orbits is a derivable partial spread containing at least two lines of $\mathrm{PG}(3,q)$. \par This result generalizes the canonical relationship between flocks of a quadratic cone and conical spreads. \par Using the properties of the elation groups, the authors prove that a deficiency 1 partial $\alpha$-flokki can be extended to an $\alpha$-flokki, generalizing the theorem of \textit{S. E. Payne} and \textit{J. A. Thas} about the deficiency 1 partial flocks of the quadratic cone [Geom. Dedicata 38, No. 2, 229--243 (1991; Zbl 0735.51010)].
scientific article; zbMATH DE number 6847108