Automorphisms mapping a point into a subvariety

TL;DR

This paper proves the undecidability of determining whether an automorphism of a complex variety can map a given point into a subvariety, and also establishes the undecidability of a version of Hilbert's tenth problem for certain polynomial systems.

Contribution

It introduces new undecidability results for automorphisms of complex varieties and for polynomial systems defining smooth projective varieties.

Findings

Decidability of automorphism mapping problem is impossible in general.

A version of Hilbert's tenth problem is undecidable for specific affine Q-varieties.

The results connect automorphism problems with classical undecidable problems.

Abstract

The problem of deciding, given a complex variety X, a point x in X, and a subvariety Z of X, whether there is an automorphism of X mapping x into Z is proved undecidable. Along the way, we prove the undecidability of a version of Hilbert's tenth problem for systems of polynomials over Z defining an affine Q-variety whose projective closure is smooth.