Let $k$ be an arbitrary field, we work with schemes of finite type over $k$. Does every irreducible projective scheme have a finite surjective morphism to $\mathbb{P}^n_k$? Certainly a $k$-scheme with this property is projective (as opposed to being proper but not projective) by formal properties of ample line bundles.

I would do this by projecting from sufficiently general points, and this probably works but I can't help but think there is a cleaner argument (that also doesn't require possibly assuming that the field is infinite, algebraically closed or of characteristic $0$).

Feel free to assume our schemes have basic niceness properties.

Let $k$ be an arbitrary field, we work with schemes $X$ of finite type over $k$. Does every irreducible projective scheme have a finite surjective morphism to a projective space $\mathbb{P}^n_k$?. What if I just assume that $X$ is equidimensional. Does the same argument work?

We know that a proper $k$-scheme with this property must be also be projective by formal properties of ample line bundles.

I would do this by projecting from sufficiently general points, and this probably works (maybe not in complete generality) but I can't help but think there is a cleaner argument (that also doesn't require possibly assuming that the field is infinite, algebraically closed or of characteristic $0$).

Feel free to assume our schemes have basic niceness properties.