Geodesic
Automorphisms of affine surfaces. II
Izvestiya. Mathematics, Tome 11 (1977) no. 1, pp. 51-98 
M. Kh. Gizatullin; V. I. Danilov. Automorphisms of affine surfaces. II. Izvestiya. Mathematics, Tome 11 (1977) no. 1, pp. 51-98. http://geodesic.mathdoc.fr/item/IM2_1977_11_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Affine surfaces $X$ completed by an irreducible rational curve $C$ are studied. The integer $m=(C^2)$ is an invariant of $X$. It is shown that the set of all such surfaces with fixed invariant $m$ is described in terms of orbits of a group action on the space of “tails”; moreover, the automorphism group $\operatorname{Aut}(X)$ is expressed by the stabilizers of the action. Explicit formulas for generators of the group $\operatorname{Aut}(X)$ are given for $m\leqslant5$. In particular, it is shown that in zero characteristic the invariant $m$ uniquely determines the surface $X$; in the general case this is not so. Bibliography: 11 titles.

[1] Gizatullin M. X., Danilov V. I., “Avtomorfizmy affinnykh poverkhnostei, I”, Izv. AN SSSR. Ser. matem., 39 (1975), 523–565 | MR

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

[3] Gizatullin M. X., “Affinnye poverkhnosti, kvaziodnorodnye otnositelno algebraicheskoi gruppy”, Izv. AN SSSR. Ser. matem., 35 (1971), 738–753 | MR

[4] Gizatullin M. X., Danilov V. I., “Primery kvaziodnorodnykh poverkhnostei”, Izv. AN SSSR. Ser. matem., 38 (1974), 42–58 | MR | Zbl

[5] Zariski O., “Polinomial ideals defined by infinitely near base points”, Amer. J. Math., 60 (1938), 151–204 | DOI | MR | Zbl

[6] Lipman J., “Rational singularities, with applications to algebraic surfaces and unique factorization”, Inst. Hautes Études Sci. Publ. Math., 1969, no. 36, 195–279 | DOI | MR | Zbl

[7] Cepp Zh.-P., “Derevya, amalgamy i $SL_2$”, Matematika, 18:1 (1974), 3–51

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

[9] Grothendieck A., Dieudonne J., “Elements de Geometrie Algebrique”, Inst. Hautes Études Sci. Publ. Math., 1961, no. 8, 222

[10] Popov V. L., “Klassifikatsiya affinnykh algebraicheskikh poverkhnostei, kvaziodnorodnykh otnositelno algebraicheskoi gruppy”, Izv. AN SSSR. Ser. matem., 37 (1973), 1038–1055 | Zbl

[11] Manin Yu. I., “Ratsionalnye poverkhnosti nad sovershennymi polyami”, Inst. Hautes Études Sci. Publ. Math., 1966, no. 30, 55–113 | DOI | MR