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We study some properties of the affine plane. First we describe the set of fixed points of a polynomial automorphism of C2. Next we classify completely so-called identity sets for polynomial automorphisms of C2. Finally, we show that a sufficiently general Zariski open affine subset of the affine plane has a finite group of automorphisms.

The automorphism groups of Zariski open affine subsets of the affine plane