Geodesic
Examples of nonhomogeneous quasihomogeneous surfaces
Izvestiya. Mathematics, Tome 8 (1974) no. 1, pp. 43-60 
M. Kh. Gizatullin; V. I. Danilov. Examples of nonhomogeneous quasihomogeneous surfaces. Izvestiya. Mathematics, Tome 8 (1974) no. 1, pp. 43-60. http://geodesic.mathdoc.fr/item/IM2_1974_8_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Over a field of arbitrary positive characteristic we construct a nonsingular affine surface $X$ which is quasihomogeneous but not homogeneous. More precisely, we find generators of the group of automorphisms of this surface and show that there exists a point $\xi\in X$ which is invariant under all the automorphisms of $X$, while $\operatorname{Aut}(X)$ acts transitively on the points of $X-\xi$.

[1] Gizatullin M. X., “Ob affinnykh poverkhnostyakh, popolnyaemykh neosoboi ratsionalnoi krivoi”, Izv. AN SSSR. Ser. matem., 34 (1970), 778–802 | MR | Zbl

[2] Gizatullin M. X., “Invarianty nepolnykh algebraicheskikh poverkhnostei, poluchaemye s pomoschyu popolnenii”, Izv. AN SSSR. Ser. matem., 35 (1971), 485–497 | MR | Zbl

[3] Gizatullin M. X., “Kvaziodnorodnye affinnye poverkhnosti”, Izv. AN SSSR. Ser. matem., 35 (1971), 1047–1071 | MR | Zbl

[4] Shafarevich I. R., Osnovy algebraicheskoi geometrii, Nauka, M., 1972 | MR | Zbl