Abstract:
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.
Fulltext:
PDF file (4518 kB)
