I am struggling with Exercise 1.5 from the book "Moduli of Curves" by Harris and Morrison. Namely, I want to show that $\mathbb{P}^1_{\mathbb{C}}$ is a fine moduli space of lines through the origin in $\mathbb{C}^2$.
Edit:
I want to phrase the question differently and more detailed. Here are my thoughts. First of all, by $\mathbb{C}^2$ I mean the affine space $\textrm{Spec} (\mathbb{C}[x,y])$. And a line through the origin is a subscheme corresponding to an ideal generated by $\lambda x+\mu y$ where $\lambda, \mu \in \mathbb{C}$ are not both zero. Now consider the functor $F$ from the category of schemes over $\mathbb{C}$ to the category of sets that sends a scheme $X$ to the set of subschemes $W \subseteq \mathbb{C}^2 \times X$ (together with the projection on $X$) that have the property that the fiber over every $\mathbb{C}$-point of $X$ is a line through the origin (the fiber is embedded to $\mathbb{C}^2$ via the projection). I want to show that $F$ is isomorphic to $\textrm{Mor}_{\mathbb{C}}(-,\mathbb{P}^1_{\mathbb{C}})$. So I want to find two natural transformations $F\to \textrm{Mor}_{\mathbb{C}}(-,\mathbb{P}^1_{\mathbb{C}})$ and $\textrm{Mor}_{\mathbb{C}}(-,\mathbb{P}^1_{\mathbb{C}})\to F$ that are inverse to each other. For the second one there is a natural candidate coming from the tautological bundle over $\mathbb{P}^1_{\mathbb{C}}$. But my problem is to show that this has an inverse: So I have a subscheme $W \subseteq \mathbb{C}^2 \times X$ that has the property that the fiber over every $\mathbb{C}$-point of $X$ is a line through the origin and I have to find a corresponding morphism $X \to \mathbb{P}^1_{\mathbb{C}}$. This means that I have to associate a line bundle to $W$ that is generated by two global sections. I guess to do that, one takes sections of the projection map $W \to X$. This gives a sheaf on $X$, but why is it coherent and locally free of rank one?