Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$? I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which is in my opinion a field, thus $(xy-1)$ is maximal. Where did I make a mistake?

The ideal $(\rm XY - 1)$ describes a curve if you draw it and over $\mathbb C$, maximal ideals describe points.
At a technical level, the answer of Martin is right.
