Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon X\hookrightarrow\bar{X}$ into a proper scheme $\bar{X}$ over $k$).

Let $\mathbf{V}/k$ be the category of separated schemes of finite type over $k$ (or any of its supercategories for which the answer of the below questions is known).

Is there known such $\mathbf{V}$ in which:

• 'minimal' compactification always exists? minimal in terms of immersions, dimension of the complement (or its cardinality over finite fields),...

• a functorial compactification is defined? I.e. an (idempotent) endofunctor $ C\colon \mathbf{V}/k\to \mathbf{V}/k $ whose image lies in the subcategory of proper 'schemes' in $\mathbf{V}/k$ and a natural morphism $\eta\colon id \Rightarrow C$ such that:

  • $\eta_X\colon X\to C(X)$ is an isomorphism for every proper $X\in \mathbf{V}/k$,
  • and (preferably) the component morphism $\eta_X\colon X\to C(X)$ is a (dense) open immersion for every $X\in \mathbf{V}/k$.

Any reference for the existence of such functorial compactification, closely related ones, or the obstruction to its existence, would be appreciated.