Geodesic
Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$
Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 535-549 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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We prove that the exceptional complex Lie group ${\mathrm{F}_4}$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of ${\mathrm{Spin}}(9,\mathbb{C})\leq {\mathrm{F}_4}$. Moreover, we identify the stabilizer of the ${\mathrm{F}_4}$-action as a parabolic subgroup ${\mathrm{P}_4}$ (with Levi factor $\mathrm{B_3T_1}$) of the complex Lie group ${\mathrm{F}_4}$. In the real case we obtain an analogous realization of ${\mathrm{F}_4}^{(-20)}/\P$.
We prove that the exceptional complex Lie group ${\mathrm{F}_4}$ has a transitive action on the hyperplane section of the complex Cayley plane ${\mathbb{O}\mathbb{P}}^2$. Although the result itself is not new, our proof is elementary and constructive. We use an explicit realization of the vector and spin actions of ${\mathrm{Spin}}(9,\mathbb{C})\leq {\mathrm{F}_4}$. Moreover, we identify the stabilizer of the ${\mathrm{F}_4}$-action as a parabolic subgroup ${\mathrm{P}_4}$ (with Levi factor $\mathrm{B_3T_1}$) of the complex Lie group ${\mathrm{F}_4}$. In the real case we obtain an analogous realization of ${\mathrm{F}_4}^{(-20)}/\P$.
Classification : 14M17, 32M12
Keywords: Cayley plane; octonionic contact structure; twistor fibration; parabolic geometry; Severi varieties; hyperplane section; exceptional geometry 
@article{CMUC_2011_52_4_a5,
     author = {Pazourek, Karel and Tu\v{c}ek, V{\'\i}t and Franek, Peter},
     title = {Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex {Cayley} plane as the homogeneous space $\mathrm{F_4/P_4}$},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {535--549},
     year = {2011},
     volume = {52},
     number = {4},
     mrnumber = {2863997},
     zbl = {1249.32019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2011_52_4_a5/}
}
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Pazourek, Karel; Tuček, Vít; Franek, Peter. Hyperplane section ${\mathbb{O}\mathbb{P}}^2_0$ of the complex Cayley plane as the homogeneous space $\mathrm{F_4/P_4}$. Commentationes Mathematicae Universitatis Carolinae, Tome 52 (2011) no. 4, pp. 535-549. http://geodesic.mathdoc.fr/item/CMUC_2011_52_4_a5/