Lemma 29.39.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that

1. the invertible sheaf $\mathcal{L}$ is ample on $X$, and

2. the morphism $X \to S$ is locally of finite type.

Then there exists a $d_0 \geq 1$ such that for every $d \geq d_0$ there exists an $n \geq 0$ and an immersion $i : X \to \mathbf{P}^ n_ S$ over $S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.

Proof. Let $A = \Gamma _*(X, \mathcal{L}) = \bigoplus _{d \geq 0} \Gamma (X, \mathcal{L}^{\otimes d})$. By Properties, Proposition 28.26.13 the set of affine opens $X_ a$ with $a \in A_{+}$ homogeneous forms a basis for the topology of $X$. Hence we can find finitely many such elements $a_0, \ldots , a_ n \in A_{+}$ such that

1. we have $X = \bigcup _{i = 0, \ldots , n} X_{a_ i}$,

2. each $X_{a_ i}$ is affine, and

3. each $X_{a_ i}$ maps into an affine open $V_ i \subset S$.

By Lemma 29.15.2 we see that the ring maps $\mathcal{O}_ S(V_ i) \to \mathcal{O}_ X(X_{a_ i})$ are of finite type. Hence we can find finitely many elements $f_{ij} \in \mathcal{O}_ X(X_{a_ i})$, $j = 1, \ldots , n_ i$ which generate $\mathcal{O}_ X(X_{a_ i})$ as an $\mathcal{O}_ S(V_ i)$-algebra. By Properties, Lemma 28.17.2 we may write each $f_{ij}$ as $a_{ij}/a_ i^{e_{ij}}$ for some $a_{ij} \in A_{+}$ homogeneous. Let $N$ be a positive integer which is a common multiple of all the degrees of the elements $a_ i$, $a_{ij}$. Consider the elements

$a_ i^{N/\deg (a_ i)}, \ a_{ij}a_ i^{(N/\deg (a_ i)) - e_{ij}} \in A_ N.$

By construction these generate the invertible sheaf $\mathcal{L}^{\otimes N}$ over $X$. Hence they give rise to a morphism

$j : X \longrightarrow \mathbf{P}_ S^{m} \quad \text{with } m = n + \sum n_ i$

over $S$, see Constructions, Lemma 27.13.1 and Definition 27.13.2. Moreover, $j^*\mathcal{O}_{\mathbf{P}_ S}(1) = \mathcal{L}^{\otimes N}$. We name the homogeneous coordinates $T_0, \ldots , T_ n, T_{ij}$ instead of $T_0, \ldots , T_ m$. For $i = 0, \ldots , n$ we have $i^{-1}(D_{+}(T_ i)) = X_{a_ i}$. Moreover, pulling back the element $T_{ij}/T_ i$ via $j^\sharp$ we get the element $f_{ij} \in \mathcal{O}_ X(X_{a_ i})$. Hence the morphism $j$ restricted to $X_{a_ i}$ gives a closed immersion of $X_{a_ i}$ into the affine open $D_{+}(T_ i) \cap \mathbf{P}^ m_{V_ i}$ of $\mathbf{P}^ N_ S$. Hence we conclude that the morphism $j$ is an immersion. This implies the lemma holds for some $d$ and $n$ which is enough in virtually all applications.

This proves that for one $d_2 \geq 1$ (namely $d_2 = N$ above), some $m \geq 0$ there exists some immersion $j : X \to \mathbf{P}^ m_ S$ given by global sections $s'_0, \ldots , s'_ m \in \Gamma (X, \mathcal{L}^{\otimes d_2})$. By Properties, Proposition 28.26.13 we know there exists an integer $d_1$ such that $\mathcal{L}^{\otimes d}$ is globally generated for all $d \geq d_1$. Set $d_0 = d_1 + d_2$. We claim that the lemma holds with this value of $d_0$. Namely, given an integer $d \geq d_0$ we may choose $s''_1, \ldots , s''_ t \in \Gamma (X, \mathcal{L}^{\otimes d - d_2})$ which generate $\mathcal{L}^{\otimes d - d_2}$ over $X$. Set $k = (m + 1)t$ and denote $s_0, \ldots , s_ k$ the collection of sections $s'_\alpha s''_\beta$, $\alpha = 0, \ldots , m$, $\beta = 1, \ldots , t$. These generate $\mathcal{L}^{\otimes d}$ over $X$ and therefore define a morphism

$i : X \longrightarrow \mathbf{P}^{k - 1}_ S$

such that $i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1) \cong \mathcal{L}^{\otimes d}$. To see that $i$ is an immersion, observe that $i$ is the composition

$X \longrightarrow \mathbf{P}^ m_ S \times _ S \mathbf{P}^{t - 1}_ S \longrightarrow \mathbf{P}^{k - 1}_ S$

where the first morphism is $(j, j')$ with $j'$ given by $s''_1, \ldots , s''_ t$ and the second morphism is the Segre embedding (Constructions, Lemma 27.13.6). Since $j$ is an immersion, so is $(j, j')$ (apply Lemma 29.3.1 to $X \to \mathbf{P}^ m_ S \times _ S \mathbf{P}^{t - 1}_ S \to \mathbf{P}^ m_ S$). Thus $i$ is a composition of immersions and hence an immersion (Schemes, Lemma 26.24.3). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01VS. Beware of the difference between the letter 'O' and the digit '0'.