Let $k$ be a field and consider the algebraic group $GL_n$ over $Spec(k)$. It has as a closed (but not normal) algebraic subgroup the group $M$ of monomial matrices, i.e. matrices having exactly one nonzero entry in each row and each column (this is the normalizer $T\rtimes\Sigma_n$ of the diagonal matrices $T$).

The geometric quotient $GL_n/M$ of the canonical action of $M$ on $GL_n$ exists (if I checked everything correctly) and it is the affine scheme associated to the ring of invariants $R$.

(Intuitively, $GL_n/M$ should be some open subset of $(\mathbb{P}^{n-1})^n$ of $n$ lines spanning the whole space with permutations identified.)

This ring of invariants $R$ is finitely generated as a $k$-algebra: $M$ is reductive, by Mumford’s Conjecture geometrically reductive and hence finitely generated by Nagata’s Theorem (it's possibly easier to see this directly in this example). I do not think, that I need an infinite field somewhere.

Is there a Zariski open covering of $GL_n/M$ by nice affine schemes $Spec(k[x_1,\ldots,x_m]/I])$ which I can explicitly write down?