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Conjugacy classes of affine automorphisms of \(\mathbb K^n\) and linear automorphisms of \(\mathbb P^n\) in the Cremona groups. (English) Zbl 1093.14017

For the algebraically closed field \(K\) of characteristic \(0\), let \(\text{Bir}(K^n)\) be the Cremona group of \(K^n\), i.e. the group of all birational isomorphisms of \(K^n\). The affine Cremona group \(\text{Aut}(n,K) \subset \text{Bir}(K^n)\) is the subgroup of all biregular automorphisms of \(K^n\), containing the group \(\text{Aff}(n,K) \supset \text{GL}(n,K)\) of affine automorphisms of \(K^n\). In this paper are described the conjugacy classes in \(\text{Aff}(n,K)\) by elements of \(\text{Aut}(n,K)\) and \(\text{Bir}(K^n)\). These descriptions take as a model the known description of the conjugacy classes of \(\text{Aut}(n,K)\): Two affine automorphisms \(\alpha, \beta \in \text{Aut}(n,K)\) are conjugate to each other by an element of \(\text{Aut}(n,K)\) if and only if they have the same Jordan normal form, see Proposition 1. Notice that this statement uses an extended understanding of a Jordan normal form that covers also the affine automorphisms with no fixed points, see page 228. If two affine automorphisms are conjugate in \(\text{Aut}(n,K)\) then either they both have fixed points or they both are without fixed points. In more detail, as shown in Section 3, two affine automorphisms with fixed points are conjugate in \(\text{Aut}(n,K)\) iff they are already conjugate in \(\text{Aff}(n,K)\). In the case when an affine automorphism has no fixed points then it is conjugate, in \(\text{Aut}(n,K)\), to an almost diagonal automorphism, see Propositions 2 and 3. In contrast, an affine automorphism with a fixed point can be conjugate in \(\text{Bir}(K^n)\) to an affine automorphism with no fixed points, see Example 3. Thus, affine automorphisms have less conjugacy classes by Cremona transformation than by only affine Cremona transformations. A rough classification is given by Proposition 4: An affine automorphism \(\alpha \in \text{Aff}(n,K)\) is conjugate in \(\text{Bir}(K^n)\) to either diagonal or to an almost diagonal affine automorphism. Next, in Section 4 is shown that the conjugacy classes in \(\text{Bir}(K^n)\) of diagonal and almost diagonal affine automorophisms are described by orbits of actions of \(\text{GL}(n,Z)\) and \(\text{GL}(n-1,Z)\). Section 5 contains similar results for the conjugacy classes of automorphisms of the the projective space \(\mathbb P^n = \mathbb P(K^{n+1})\) by elements of the Cremona group \(\text{Bir}(\mathbb P^n) \cong \text{Bir}(K^n)\).

MSC:

14E07 Birational automorphisms, Cremona group and generalizations
14J50 Automorphisms of surfaces and higher-dimensional varieties
14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
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References:

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