On the set of fixed points of a polynomial automorphism

Abstract

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