On the set of fixed points of a polynomial automorphism

TL;DR

This paper investigates the geometric structure of fixed points of special and non-special polynomial automorphisms over algebraically closed fields, revealing uniruledness and smoothness properties of fixed point components.

Contribution

It proves that fixed point components of special automorphisms are uniruled, and characterizes fixed points of non-special automorphisms as smooth, irreducible, and with Euler characteristic 1 over complex numbers.

Findings

(n-1)-dimensional fixed point components are uniruled for special automorphisms

Fixed points of non-special automorphisms are smooth and irreducible

Euler characteristic of fixed points is 1 over complex numbers

Abstract

Let K be an algebraically closed field of characteristic zero. We say that a polynomial automorphism f : K^n -> 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 : K^n -> 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) and for K = C the Euler characteristic of H is equal to 1.