Is $\mathbb{P}^{1}$ a fine moduli scheme?

Question

I want to show that $\mathbb{P}^{1}_{\mathbb{C}}$ is a fine moduli scheme for the families of lines through the origin of the affine plane. I took a flat family $\mathcal{D}\rightarrow B$ and I tried to associate to it a morphism $B\rightarrow \mathbb{P}^{1}_{\mathbb{C}}$ in order to prove that the moduli functor is representable, but right now I'm a little bit stuck because of the big generality (almost for me) of the request. Can someone give me some help? Thank you in advance!

Answer 1

$\def\D{{\mathcal D}} \def\A{{\mathbb A}} \def\L{{\mathcal L}} \def\OO{{\mathcal O}}$Let $\pi: \D \to B$ be a family of lines through the origin in the plane. So $\D \subset B \times \A^2$ with $\pi$ the restriction of the natural projection. One can prove that such a family is actually a line bundle. Here the fact that the fibers are each naturally vector subspaces of $\A^2$ and not just isomorphic to lines is important.

Then the correspondence between vector bundles and locally free sheaves gives us an invertible sheaf $\L$ on $B$ and the embedding $\D \subset B \times \A^2$ gives an embedding of locally free sheaves

$$ 0 \to \L \to \OO_B^{\oplus 2} $$

into the rank 2 free sheaf. Taking duals gives us a surjective morphism of sheaves

$$ \OO_B^{\oplus 2} \to \L^\vee \to 0 \enspace \enspace \enspace (*) $$

A morphism $\OO_B \to \mathcal{F}$ for any sheaf $\mathcal{F}$ on $B$ is just a global section of $\mathcal{F}$. So $(*)$ tells us that $\L^\vee$ is globally generated by two global sections, say $s_1$ and $s_2$ (i.e. the images of the generators of the free sheaf in $\L^\vee$).

Then we get a natural morphism $\varphi: B \to \mathbb{P}^1$ given by $b \mapsto [s_1(b):s_2(b)]$ so that $\varphi^*(\OO(1),x_1,x_2) = (\L^\vee,s_1,s_2)$ where $x_i$ are the sections of $\OO(1)$ coming from coordinates on $\A^2$. Dualizing we get that $\varphi^*\OO(-1) = \L$ or on the level of line bundles instead of sheaves, $\varphi^*\mathcal{T} = \mathcal{D}$ where $\mathcal{T}$ is the tautological bundle on $\mathbb{P}^1$.