## 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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