The Stacks project

Example 27.21.2 (Projective space of a vector space). Let $k$ be a field. Let $V$ be a $k$-vector space. The corresponding projective space is the $k$-scheme

\[ \mathbf{P}(V) = \text{Proj}(\text{Sym}(V)) \]

where $\text{Sym}(V)$ is the symmetric algebra on $V$ over $k$. Of course we have $\mathbf{P}(V) \cong \mathbf{P}^ n_ k$ if $\dim (V) = n + 1$ because then the symmetric algebra on $V$ is isomorphic to a polynomial ring in $n + 1$ variables. If we think of $V$ as a quasi-coherent module on $\mathop{\mathrm{Spec}}(k)$, then $\mathbf{P}(V)$ is the corresponding projective space bundle over $\mathop{\mathrm{Spec}}(k)$. By the discussion above a $k$-valued point $p$ of $\mathbf{P}(V)$ corresponds to a surjection of $k$-vector spaces $V \to L_ p$ with $\dim (L_ p) = 1$. More generally, let $X$ be a scheme over $k$, let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module, and let $\psi : V \to \Gamma (X, \mathcal{L})$ be a $k$-linear map such that $\mathcal{L}$ is generated as an $\mathcal{O}_ X$-module by the sections in the image of $\psi $. Then the discussion above gives a canonical morphism

\[ \varphi _{\mathcal{L}, \psi } : X \longrightarrow \mathbf{P}(V) \]

of schemes over $k$ such that there is an isomorphism $\theta : \varphi _{\mathcal{L}, \psi }^*\mathcal{O}_{\mathbf{P}(V)}(1) \to \mathcal{L}$ and such that $\psi $ agrees with the composition

\[ V \to \Gamma (\mathbf{P}(V), \mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma (X, \varphi _{\mathcal{L}, \psi }^*\mathcal{O}_{\mathbf{P}(V)}(1)) \to \Gamma (X, \mathcal{L}) \]

See Lemma 27.14.1. If $V \subset \Gamma (X, \mathcal{L})$ is a subspace, then we will denote the morphism constructed above simply as $\varphi _{\mathcal{L}, V}$. If $\dim (V) = n + 1$ and we choose a basis $v_0, \ldots , v_ n$ of $V$ then the diagram

\[ \xymatrix{ X \ar@{=}[d] \ar[rr]_{\varphi _{\mathcal{L}, \psi }} & & \mathbf{P}(V) \ar[d]^{\cong } \\ X \ar[rr]^{\varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}} & & \mathbf{P}^ n_ k } \]

is commutative, where $s_ i = \psi (v_ i) \in \Gamma (X, \mathcal{L})$, where $\varphi _{(\mathcal{L}, (s_0, \ldots , s_ n))}$ is as in Section 27.13, and where the right vertical arrow corresponds to the isomorphism $k[T_0, \ldots , T_ n] \to \text{Sym}(V)$ sending $T_ i$ to $v_ i$.

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