The Stacks project

Exercise 111.24.4. Let $k$ be an algebraically closed field. Compute the image in $\mathop{\mathrm{Spec}}(k[x, y])$ of the following maps:

1. $\mathop{\mathrm{Spec}}(k[x, yx^{-1}]) \to \mathop{\mathrm{Spec}}(k[x, y])$, where $k[x, y] \subset k[x, yx^{-1}] \subset k[x, y, x^{-1}]$. (Hint: To avoid confusion, give the element $yx^{-1}$ another name.)

2. $\mathop{\mathrm{Spec}}(k[x, y, a, b]/(ax-by-1))\to \mathop{\mathrm{Spec}}(k[x, y])$.

3. $\mathop{\mathrm{Spec}}(k[t, 1/(t-1)]) \to \mathop{\mathrm{Spec}}(k[x, y])$, induced by $x \mapsto t^2$, and $y \mapsto t^3$.

4. $k = {\mathbf C}$ (complex numbers), $\mathop{\mathrm{Spec}}(k[s, t]/(s^3 + t^3-1)) \to \mathop{\mathrm{Spec}}(k[x, y])$, where $x\mapsto s^2$, $y \mapsto t^2$.