Lemma 29.39.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume that
the invertible sheaf $\mathcal{L}$ is very ample on $X/S$,
the morphism $X \to S$ is of finite type, and
$S$ is affine.
29.39 Ample and very ample sheaves relative to finite type morphisms
In fact most of the material in this section is about the notion of a (quasi-)projective morphism which we have not defined yet.
Then there exist an $n \geq 0$ and an immersion $i : X \to \mathbf{P}^ n_ S$ over $S$ such that $\mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.
Proof. Assume (1), (2) and (3). Condition (3) means $S = \mathop{\mathrm{Spec}}(R)$ for some ring $R$. Condition (1) means by definition there exists a quasi-coherent $\mathcal{O}_ S$-module $\mathcal{E}$ and an immersion $\alpha : X \to \mathbf{P}(\mathcal{E})$ such that $\mathcal{L} = \alpha ^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1)$. Write $\mathcal{E} = \widetilde{M}$ for some $R$-module $M$. Thus we have
\[ \mathbf{P}(\mathcal{E}) = \text{Proj}(\text{Sym}_ R(M)). \]
Since $\alpha $ is an immersion, and since the topology of $\text{Proj}(\text{Sym}_ R(M))$ is generated by the standard opens $D_{+}(f)$, $f \in \text{Sym}_ R^ d(M)$, $d \geq 1$, we can find for each $x \in X$ an $f \in \text{Sym}_ R^ d(M)$, $d \geq 1$, with $\alpha (x) \in D_{+}(f)$ such that
\[ \alpha |_{\alpha ^{-1}(D_{+}(f))} : \alpha ^{-1}(D_{+}(f)) \to D_{+}(f) \]
is a closed immersion. Condition (2) implies $X$ is quasi-compact. Hence we can find a finite collection of elements $f_ j \in \text{Sym}_ R^{d_ j}(M)$, $d_ j \geq 1$ such that for each $f = f_ j$ the displayed map above is a closed immersion and such that $\alpha (X) \subset \bigcup D_{+}(f_ j)$. Write $U_ j = \alpha ^{-1}(D_{+}(f_ j))$. Note that $U_ j$ is affine as a closed subscheme of the affine scheme $D_{+}(f_ j)$. Write $U_ j = \mathop{\mathrm{Spec}}(A_ j)$. Condition (2) also implies that $A_ j$ is of finite type over $R$, see Lemma 29.15.2. Choose finitely many $x_{j, k} \in A_ j$ which generate $A_ j$ as a $R$-algebra. Since $\alpha |_{U_ j}$ is a closed immersion we see that $x_{j, k}$ is the image of an element
\[ f_{j, k}/f_ j^{e_{j, k}} \in \text{Sym}_ R(M)_{(f_ j)} = \Gamma (D_{+}(f_ j), \mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}). \]
Finally, choose $n \geq 1$ and elements $y_0, \ldots , y_ n \in M$ such that each of the polynomials $f_ j, f_{j, k} \in \text{Sym}_ R(M)$ is a polynomial in the elements $y_ t$ with coefficients in $R$. Consider the graded ring map
\[ \psi : R[Y_0, \ldots , Y_ n] \longrightarrow \text{Sym}_ R(M), \quad Y_ i \longmapsto y_ i. \]
Denote $F_ j$, $F_{j, k}$ the elements of $R[Y_0, \ldots , Y_ n]$ such that $\psi (F_ j) = f_ j$ and $\psi (F_{j, k}) = f_{j, k}$. By Constructions, Lemma 27.11.1 we obtain an open subscheme
\[ U(\psi ) \subset \text{Proj}(\text{Sym}_ R(M)) \]
and a morphism $r_\psi : U(\psi ) \to \mathbf{P}^ n_ R$. This morphism satisfies $r_\psi ^{-1}(D_{+}(F_ j)) = D_{+}(f_ j)$, and hence we see that $\alpha (X) \subset U(\psi )$. Moreover, it is clear that
\[ i = r_\psi \circ \alpha : X \longrightarrow \mathbf{P}^ n_ R \]
is still an immersion since $i^\sharp (F_{j, k}/F_ j^{e_{j, k}}) = x_{j, k} \in A_ j = \Gamma (U_ j, \mathcal{O}_ X)$ by construction. Moreover, the morphism $r_\psi $ comes equipped with a map $\theta : r_\psi ^*\mathcal{O}_{\mathbf{P}^ n_ R}(1) \to \mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}(1)|_{U(\psi )}$ which is an isomorphism in this case (for construction $\theta $ see lemma cited above; some details omitted). Since the original map $\alpha $ was assumed to have the property that $\mathcal{L} = \alpha ^*\mathcal{O}_{\text{Proj}(\text{Sym}_ R(M))}(1)$ we win. $\square$
Lemma 29.39.2. Let $\pi : X \to S$ be a morphism of schemes. Assume that $X$ is quasi-affine and that $\pi $ is locally of finite type. Then there exist $n \geq 0$ and an immersion $i : X \to \mathbf{A}^ n_ S$ over $S$.
Proof. Let $A = \Gamma (X, \mathcal{O}_ X)$. By assumption $X$ is quasi-compact and is identified with an open subscheme of $\mathop{\mathrm{Spec}}(A)$, see Properties, Lemma 28.18.4. Moreover, the set of opens $X_ f$, for those $f \in A$ such that $X_ f$ is affine, forms a basis for the topology of $X$, see the proof of Properties, Lemma 28.18.4. Hence we can find a finite number of $f_ j \in A$, $j = 1, \ldots , m$ such that $X = \bigcup X_{f_ j}$, and such that $\pi (X_{f_ j}) \subset V_ j$ for some affine open $V_ j \subset S$. By Lemma 29.15.2 the ring maps $\mathcal{O}(V_ j) \to \mathcal{O}(X_{f_ j}) = A_{f_ j}$ are of finite type. Thus we may choose $a_1, \ldots , a_ N \in A$ such that the elements $a_1, \ldots , a_ N, 1/f_ j$ generate $A_{f_ j}$ over $\mathcal{O}(V_ j)$ for each $j$. Take $n = m + N$ and let
\[ i : X \longrightarrow \mathbf{A}^ n_ S \]
be the morphism given by the global sections $f_1, \ldots , f_ m, a_1, \ldots , a_ N$ of the structure sheaf of $X$. Let $D(x_ j) \subset \mathbf{A}^ n_ S$ be the open subscheme where the $j$th coordinate function is nonzero. Then for $1 \leq j \leq m$ we have $i^{-1}(D(x_ j)) = X_{f_ j}$ and the induced morphism $X_{f_ j} \to D(x_ j)$ factors through the affine open $\mathop{\mathrm{Spec}}(\mathcal{O}(V_ j)[x_1, \ldots , x_ n, 1/x_ j])$ of $D(x_ j)$. Since the ring map $\mathcal{O}(V_ j)[x_1, \ldots , x_ n, 1/x_ j] \to A_{f_ j}$ is surjective by construction we conclude that $i^{-1}(D(x_ j)) \to D(x_ j)$ is an immersion as desired. $\square$
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
the invertible sheaf $\mathcal{L}$ is ample on $X$, and
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
we have $X = \bigcup _{i = 0, \ldots , n} X_{a_ i}$,
each $X_{a_ i}$ is affine, and
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). $\square$
Lemma 29.39.4. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ affine and $f$ of finite type. The following are equivalent
$\mathcal{L}$ is ample on $X$,
$\mathcal{L}$ is $f$-ample,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$,
for some $d \geq 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$, and
for all $d \gg 1$ there exist $n \geq 1$ and an immersion $i : X \to \mathbf{P}^ n_ S$ such that $\mathcal{L}^{\otimes d} \cong i^*\mathcal{O}_{\mathbf{P}^ n_ S}(1)$.
Proof. The equivalence of (1) and (2) is Lemma 29.37.5. The implication (2) $\Rightarrow $ (6) is Lemma 29.39.3. Trivially (6) implies (5). As $\mathbf{P}^ n_ S$ is a projective bundle over $S$ (see Constructions, Lemma 27.21.5) we see that (5) implies (3) and (6) implies (4) from the definition of a relatively very ample sheaf. Trivially (4) implies (3). To finish we have to show that (3) implies (2) which follows from Lemma 29.38.2 and Lemma 29.37.2. $\square$
Lemma 29.39.5. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Assume $S$ quasi-compact and $f$ of finite type. The following are equivalent
$\mathcal{L}$ is $f$-ample,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for some $d \geq 1$,
$\mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$.
Proof. Trivially (3) implies (2). Lemma 29.38.2 guarantees that (2) implies (1) since a morphism of finite type is quasi-compact by definition. Assume that $\mathcal{L}$ is $f$-ample. Choose a finite affine open covering $S = V_1 \cup \ldots \cup V_ m$. Write $X_ i = f^{-1}(V_ i)$. By Lemma 29.39.4 above we see there exists a $d_0$ such that $\mathcal{L}^{\otimes d}$ is relatively very ample on $X_ i/V_ i$ for all $d \geq d_0$. Hence we conclude (1) implies (3) by Lemma 29.38.7. $\square$
The following two lemmas provide the most used and most useful characterizations of relatively very ample and relatively ample invertible sheaves when the morphism is of finite type.
Lemma 29.39.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent:
$\mathcal{L}$ is $f$-relatively very ample, and
there exist an open covering $S = \bigcup V_ j$, for each $j$ an integer $n_ j$, and immersions
over $V_ j$ such that $\mathcal{L}|_{X_ j} \cong i_ j^*\mathcal{O}_{\mathbf{P}^{n_ j}_{V_ j}}(1)$.
\[ i_ j : X_ j = f^{-1}(V_ j) = V_ j \times _ S X \longrightarrow \mathbf{P}^{n_ j}_{V_ j} \]
Proof. We see that (1) implies (2) by taking an affine open covering of $S$ and applying Lemma 29.39.1 to each of the restrictions of $f$ and $\mathcal{L}$. We see that (2) implies (1) by Lemma 29.38.7. $\square$
Lemma 29.39.7. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{L}$ be an invertible sheaf on $X$. Assume $f$ is of finite type. The following are equivalent:
$\mathcal{L}$ is $f$-relatively ample, and
there exist an open covering $S = \bigcup V_ j$, for each $j$ an integers $d_ j \geq 1$, $n_ j \geq 0$, and immersions
over $V_ j$ such that $\mathcal{L}^{\otimes d_ j}|_{X_ j} \cong i_ j^*\mathcal{O}_{\mathbf{P}^{n_ j}_{V_ j}}(1)$.
\[ i_ j : X_ j = f^{-1}(V_ j) = V_ j \times _ S X \longrightarrow \mathbf{P}^{n_ j}_{V_ j} \]
Proof. We see that (1) implies (2) by taking an affine open covering of $S$ and applying Lemma 29.39.4 to each of the restrictions of $f$ and $\mathcal{L}$. We see that (2) implies (1) by Lemma 29.37.4. $\square$
Lemma 29.39.8. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{N}$, $\mathcal{L}$ be invertible $\mathcal{O}_ X$-modules. Assume $S$ is quasi-compact, $f$ is of finite type, and $\mathcal{L}$ is $f$-ample. Then $\mathcal{N} \otimes _{\mathcal{O}_ X} \mathcal{L}^{\otimes d}$ is $f$-very ample for all $d \gg 1$.
Proof. By Lemma 29.39.6 we reduce to the case $S$ is affine. Combining Lemma 29.39.4 and Properties, Proposition 28.26.13 we can find an integer $d_0$ such that $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ is globally generated. Choose global sections $s_0, \ldots , s_ n$ of $\mathcal{N} \otimes \mathcal{L}^{\otimes d_0}$ which generate it. This determines a morphism $j : X \to \mathbf{P}^ n_ S$ over $S$. By Lemma 29.39.4 we can also pick an integer $d_1$ such that for all $d \geq d_1$ there exist sections $t_{d, 0}, \ldots , t_{d, n(d)}$ of $\mathcal{L}^{\otimes d}$ which generate it and define an immersion
\[ j_ d = \varphi _{\mathcal{L}^{\otimes d}, t_{d, 0}, \ldots , t_{d, n(d)}} : X \longrightarrow \mathbf{P}^{n(d)}_ S \]
over $S$. Then for $d \geq d_0 + d_1$ we can consider the morphism
\[ \varphi _{\mathcal{N} \otimes \mathcal{L}^{\otimes d}, s_ j \otimes t_{d - d_0, i}} : X \longrightarrow \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_ S \]
This morphism is an immersion as it is the composition
\[ X \to \mathbf{P}^ n_ S \times _ S \mathbf{P}^{n(d - d_0)}_ S \to \mathbf{P}^{(n + 1)(n(d - d_0) + 1) - 1}_ S \]
where the first morphism is $(j, j_{d - d_0})$ and the second is the Segre embedding (Constructions, Lemma 27.13.6). Since $j$ is an immersion, so is $(j, j_{d - d_0})$ (apply Lemma 29.3.1). We have a composition of immersions and hence an immersion (Schemes, Lemma 26.24.3). $\square$
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