TITLE: Total space of the line bundle $\mathcal{O}(1)$ over $\mathbb{P}^n$ 
QUESTION [22 upvotes]: It is well known that total space of the tautological line bundle $\mathcal{O}(-1)$ over projective space $\mathbb{P}^n$ is closed subvariety of $\mathbb{P}^n\times\mathbb{A}^{n+1}$. My question is how to realize total space of $\mathcal{O}(1)$ over $\mathbb{P}^n$ in such manner, i.e. I need an embedding of $Tot(\mathcal{O}(1))$ in simple variety and defining equations. 
Thanks.

REPLY [11 votes]: For me the best description of $Tot(O(1))$ is the tautological --- as the relative spectrum of the sheaf of algebras $A = O \oplus O(-1) \oplus O(-2) \oplus \dots$ on $P^n$:
$$
Tot(O(1)) = Spec_{P^n}(A).
$$
This allows to work with $Tot(O(1))$ more effectively than any other description. For example, a coherent sheaf on $Tot(O(1))$ can be represented by a quasicoherent sheaf $F$ on $P^n$ together with a morphism $F(-1) \to F$ inducing on $F$ a structure of an $A$-module.