Images of dominant endomorphisms of affine space

A basic problem in the study of algebraic morphisms is to determine which sets can be realised as the image of an endomorphism of affine space. This paper extends the results previously obtained by the first author on the question of existence of surjective maps $F\colon \mathbb{A}^n \rightarrow \mathbb{A}^n\setminus Z$, where $Z$ is an algebraic subvariety of $\mathbb{A}^n$ of codimension at least 2. In particular, we show that for any (affine) algebraic variety $Z$ of dimension at most $n-2$, there is an algebraic variety $W\subset \mathbb{A}^n$ birational to $Z$ and a surjective algebraic morphism $\mathbb{A}^n\rightarrow \mathbb{A}^n\setminus W$. We also propose a conjectural approach towards resolving unknown cases.