= Determinant-one cyclic quotient of the affine plane
{title2=$\mathbb C[x,y]^{\mu_n}\cong\mathbb C[u,v,w]/(uv-w^n)$}
The action $(x,y)\mapsto(\zeta x,\zeta^{-1}y)$ has invariant <monomials> exactly when their two exponents differ by a multiple of $n$. Every such <monomial> is a product of $x^n,y^n,xy$. Reducing by $uv=w^n$ leaves the <monomials> $w^j$, $u^iw^j$ and $v^iw^j$ with $i\ge1$; their images are distinct <monomials> in $x,y$, proving that this is the only relation. The map $(x,y)\mapsto(x^n,y^n,xy)$ is finite and its fibres are the finite-group orbits, so the hypersurface is the <geometric quotient>.
// Destination: ringed-space.bigb
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