The Stacks project

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 exist 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). $\square$

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