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On the set of fixed points of a polynomial automorphism. (English) Zbl 1349.14191

Summary: Let \(\mathbb{K}\) be an algebraically closed field of characteristic zero. We say that a polynomial automorphism \(f:\mathbb{K}^n\to\mathbb{K}^n\) is special if the Jacobian of \(f\) is equal to 1. We show that every \((n-1)\)-dimensional component \(H\) of the set \(\mathrm{Fix}(f)\) of fixed points of a non-trivial special polynomial automorphism \(f:\mathbb{K}^n\to\mathbb{K}^n\) is uniruled. Moreover, we show that if \(f\) is non-special and \(H\) is an \((n-1)\)-dimensional component of the set \(\mathrm{Fix}(f)\), then \(H\) is smooth, irreducible and \(H=\mathrm{Fix}(f)\). Moreover, for \(\mathbb{K}=\mathbb{C}\) if \(f\) is non-special and \(\mathrm{Jac}(f)\) has an infinite order in \(\mathbb{C}^\ast\), then the Euler characteristic of \(H\) is equal to 1.

MSC:

14R10 Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem)
14R20 Group actions on affine varieties
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