Sheaves and Algebraic Geometry

Let $k$ be a field of characteristic zero (just to be safe) and define the projective scheme $$X = \operatorname{Proj} k[x,y,z]/(xy-z^2)$$ over $\operatorname{Spec}k$. Since $X$ is a conic with a rational point, it is isomorphic to $\mathbb{P}^1 = \mathbb{P}^1_k$ by the "slope-parameterization" trick.

Consider the following alternative argument for constructing an isomorphism.

• Step 1: The line bundle (invertible sheaf) $\mathcal{O}_X(1)$ is globally generated (on stalks) by $x,y\in \Gamma(X,\mathcal{O}_X(1))$.
• Step 2: Therefore, $x,y$ define a morphism of schemes $f\colon X \to \mathbb{P}^1$ in the usual way, informally given by the formula $[x:y]$.
• Step 3: Because $f^\ast \mathcal{O}_{ \mathbb{P}^1}(1) \cong \mathcal{O}_X(1)$, the morphism $f$ is in fact an isomorphism of schemes.