# Sections of projective bundles

Suppose we have a vector bundle $\pi:\text{Spec}~A[x_1,\cdots,x_n] \rightarrow \text{Spec}~A$, then the sections are morphisms $s:\text{Spec}~A \rightarrow \text{Spec}~A[x_1,\cdots,x_n]$ such that $\pi \circ s=\text{Identity}$, which are just $A$ homomorphisms $$A[x_1,\cdots,x_n] \rightarrow A$$ So the sections are naturally isomorphic to $A^n$. So what about the projective case? Suppose we have a projective bundle $\text{Proj}~A[x_0,x_1,\cdots,x_n] \rightarrow \text{Spec}~A$, how to find all the sections $s:\text{Spec}~A \rightarrow \text{Proj}~A[x_0,x_1,\cdots,x_n]$ ?

A section $s$ is just an $A$-morphism from $\text{Spec}~A$ to $\text{Proj}~A[x_0,x_1,\cdots,x_n]$, which corresponds to a line bundle $\mathcal{L}$ and $n+1$ sections $a_0,a_1,\cdots,a_n$ that generate $\mathcal{L}$ at every point of $\text{Spec}~A$ up to isomorphism. Since $\text{Spec}~A$ is trivially affine, $\mathcal{L}$ is trivial and isomorphic to $\mathcal{O}_{\text{Spec}A}$, thus the $n+1$ sections are just elements of $A$. So sections are in one to one bijection with the data $n+1$ elements of $A$, $(a_0,a_1,\cdots,a_n)$ that for every $[p] \in \text{Spec}A$, there is an $a_i$ that does not vanish at $[p]$, up to isomorphism. The isomorphism is given by $(a_0,a_1,\cdots,a_n) \sim (f a_0,fa_1,\cdots,fa_n)$, where $f\in A$ is a unit.