by S.Yu. Orevkov

We give a simple proof to the following result proved earlier by Vitushkin. Let us fix a line   L   on   CP²   (the infinite line of the affine plane). Let   V   be obtained from   CP²   by a chain of blowups at infinity, i.e. only points of   L   and their infinitly close points are blown up. Let   E   be the exceptional curve of the last blowup. Denote by   S   the dual homological class of   E , i.e.   S.E = 1   and   S.D = 0   for any other exceptional curve   D . We say that the chain of blowups defines an authomorphism of   C²  if all the exceptional curves except   E   can be blown down so that the obtained surface is again  CP² .

Theorem. The chain of blowups defines an authomorphism of   C²   if and only if   S.S = 1   and   S.K = -3   where   K   is the canonical class of   V .