Definition 29.38.1. Let f : X \to S be a morphism of schemes. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. We say \mathcal{L} is relatively very ample or more precisely f-relatively very ample, or very ample on X/S, or f-very ample if there exist a quasi-coherent \mathcal{O}_ S-module \mathcal{E} and an immersion i : X \to \mathbf{P}(\mathcal{E}) over S such that \mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1).
29.38 Very ample sheaves
Recall that given a quasi-coherent sheaf \mathcal{E} on a scheme S the projective bundle associated to \mathcal{E} is the morphism \mathbf{P}(\mathcal{E}) \to S, where \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\text{Sym}(\mathcal{E})), see Constructions, Definition 27.21.1.
Since there is no assumption of quasi-compactness in this definition it is not true in general that a relatively very ample invertible sheaf is a relatively ample invertible sheaf.
Lemma 29.38.2.reference Let f : X \to S be a morphism of schemes. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. If f is quasi-compact and \mathcal{L} is a relatively very ample invertible sheaf, then \mathcal{L} is a relatively ample invertible sheaf.
Proof. By definition there exists quasi-coherent \mathcal{O}_ S-module \mathcal{E} and an immersion i : X \to \mathbf{P}(\mathcal{E}) over S such that \mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1). Set \mathcal{A} = \text{Sym}(\mathcal{E}), so \mathbf{P}(\mathcal{E}) = \underline{\text{Proj}}_ S(\mathcal{A}) by definition. The graded \mathcal{O}_ S-algebra \mathcal{A} comes equipped with a map
\psi : \mathcal{A} \to \bigoplus \nolimits _{n \geq 0} \pi _*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(n) \to \bigoplus \nolimits _{n \geq 0} f_*\mathcal{L}^{\otimes n}
where the second arrow uses the identification \mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1). By adjointness of f_* and f^* we get a morphism \psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0}\mathcal{L}^{\otimes n}. We omit the verification that the morphism r_{\mathcal{L}, \psi } associated to this map is exactly the immersion i. Hence the result follows from part (6) of Lemma 29.37.4. \square
To arrive at the correct converse of this lemma we ask whether given a relatively ample invertible sheaf \mathcal{L} there exists an integer n \geq 1 such that \mathcal{L}^{\otimes n} is relatively very ample? In general this is false. There are several things that prevent this from being true:
Even if S is affine, it can happen that no finite integer n works because X \to S is not of finite type, see Example 29.38.4.
The base not being quasi-compact means the result can be prevented from being true even with f finite type. Namely, given a field k there exists a scheme X_ d of finite type over k with an ample invertible sheaf \mathcal{O}_{X_ d}(1) so that the smallest tensor power of \mathcal{O}_{X_ d}(1) which is very ample is the dth power. See Example 29.38.5. Taking f to be the disjoint union of the schemes X_ d mapping to the disjoint union of copies of \mathop{\mathrm{Spec}}(k) gives an example.
To see our version of the converse take a look at Lemma 29.39.5 below. We will do some preliminary work before proving it.
Example 29.38.3. Let S be a scheme. Let \mathcal{A} be a quasi-coherent graded \mathcal{O}_ S-algebra generated by \mathcal{A}_1 over \mathcal{A}_0. Set X = \underline{\text{Proj}}_ S(\mathcal{A}). In this case \mathcal{O}_ X(1) is a very ample invertible sheaf on X. Namely, the morphism associated to the graded \mathcal{O}_ S-algebra map
\text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A}
is a closed immersion X \to \mathbf{P}(\mathcal{A}_1) which pulls back \mathcal{O}_{\mathbf{P}(\mathcal{A}_1)}(1) to \mathcal{O}_ X(1), see Constructions, Lemma 27.18.5.
Example 29.38.4. Let k be a field. Consider the graded k-algebra
A = k[U, V, Z_1, Z_2, Z_3, \ldots ]/I \quad \text{with} \quad I = (U^2 - Z_1^2, U^4 - Z_2^2, U^6 - Z_3^2, \ldots )
with grading given by \deg (U) = \deg (V) = \deg (Z_1) = 1 and \deg (Z_ d) = d. Note that X = \text{Proj}(A) is covered by D_{+}(U) and D_{+}(V). Hence the sheaves \mathcal{O}_ X(n) are all invertible and isomorphic to \mathcal{O}_ X(1)^{\otimes n}. In particular \mathcal{O}_ X(1) is ample and f-ample for the morphism f : X \to \mathop{\mathrm{Spec}}(k). We claim that no power of \mathcal{O}_ X(1) is f-relatively very ample. Namely, it is easy to see that \Gamma (X, \mathcal{O}_ X(n)) is the degree n summand of the algebra A. Hence if \mathcal{O}_ X(n) were very ample, then X would be a closed subscheme of a projective space over k and hence of finite type over k. On the other hand D_{+}(V) is the spectrum of k[t, t_1, t_2, \ldots ]/(t^2 - t_1^2, t^4 - t_2^2, t^6 - t_3^2, \ldots ) which is not of finite type over k.
Example 29.38.5. Let k be an infinite field. Let \lambda _1, \lambda _2, \lambda _3, \ldots be pairwise distinct elements of k^*. (This is not strictly necessary, and in fact the example works perfectly well even if all \lambda _ i are equal to 1.) Consider the graded k-algebra
A_ d = k[U, V, Z]/I_ d \quad \text{with} \quad I_ d = (Z^2 - \prod \nolimits _{i = 1}^{2d} (U - \lambda _ i V)).
with grading given by \deg (U) = \deg (V) = 1 and \deg (Z) = d. Then X_ d = \text{Proj}(A_ d) has ample invertible sheaf \mathcal{O}_{X_ d}(1). We claim that if \mathcal{O}_{X_ d}(n) is very ample, then n \geq d. The reason for this is that Z has degree d, and hence \Gamma (X_ d, \mathcal{O}_{X_ d}(n)) = k[U, V]_ n for n < d. Details omitted.
Lemma 29.38.6. Let f : X \to S be a morphism of schemes. Let \mathcal{L} be an invertible sheaf on X. If \mathcal{L} is relatively very ample on X/S then f is separated.
Proof. Being separated is local on the base (see Schemes, Section 26.21). An immersion is separated (see Schemes, Lemma 26.23.8). Hence the lemma follows since locally X has an immersion into the homogeneous spectrum of a graded ring which is separated, see Constructions, Lemma 27.8.8. \square
Lemma 29.38.7. Let f : X \to S be a morphism of schemes. Let \mathcal{L} be an invertible sheaf on X. Assume f is quasi-compact. The following are equivalent
\mathcal{L} is relatively very ample on X/S,
there exists an open covering S = \bigcup V_ j such that \mathcal{L}|_{f^{-1}(V_ j)} is relatively very ample on f^{-1}(V_ j)/V_ j for all j,
there exists a quasi-coherent sheaf of graded \mathcal{O}_ S-algebras \mathcal{A} generated in degree 1 over \mathcal{O}_ S and a map of graded \mathcal{O}_ X-algebras \psi : f^*\mathcal{A} \to \bigoplus _{n \geq 0} \mathcal{L}^{\otimes n} such that f^*\mathcal{A}_1 \to \mathcal{L} is surjective and the associated morphism r_{\mathcal{L}, \psi } : X \to \underline{\text{Proj}}_ S(\mathcal{A}) is an immersion, and
f is quasi-separated, the canonical map \psi : f^*f_*\mathcal{L} \to \mathcal{L} is surjective, and the associated map r_{\mathcal{L}, \psi } : X \to \mathbf{P}(f_*\mathcal{L}) is an immersion.
Proof. It is clear that (1) implies (2). It is also clear that (4) implies (1); the hypothesis of quasi-separation in (4) is used to guarantee that f_*\mathcal{L} is quasi-coherent via Schemes, Lemma 26.24.1.
Assume (2). We will prove (4). Let S = \bigcup V_ j be an open covering as in (2). Set X_ j = f^{-1}(V_ j) and f_ j : X_ j \to V_ j the restriction of f. We see that f is separated by Lemma 29.38.6 (as being separated is local on the base). By assumption there exists a quasi-coherent \mathcal{O}_{V_ j}-module \mathcal{E}_ j and an immersion i_ j : X_ j \to \mathbf{P}(\mathcal{E}_ j) with \mathcal{L}|_{X_ j} \cong i_ j^*\mathcal{O}_{\mathbf{P}(\mathcal{E}_ j)}(1). The morphism i_ j corresponds to a surjection f_ j^*\mathcal{E}_ j \to \mathcal{L}|_{X_ j}, see Constructions, Section 27.21. This map is adjoint to a map \mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j} such that the composition
f_ j^*\mathcal{E}_ j \to (f^*f_*\mathcal{L})|_{X_ j} \to \mathcal{L}|_{X_ j}
is surjective. We conclude that \psi : f^*f_*\mathcal{L} \to \mathcal{L} is surjective. Let r_{\mathcal{L}, \psi } : X \to \mathbf{P}(f_*\mathcal{L}) be the associated morphism. We still have to show that r_{\mathcal{L}, \psi } is an immersion; we urge the reader to prove this for themselves. The \mathcal{O}_{V_ j}-module map \mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j} determines a homomorphism on symmetric algebras, which in turn defines a morphism
\mathbf{P}(f_*\mathcal{L}|_{V_ j}) \supset U_ j \longrightarrow \mathbf{P}(\mathcal{E}_ j)
where U_ j is the open subscheme of Constructions, Lemma 27.18.1. The compatibility of \psi with \mathcal{E}_ j \to f_*\mathcal{L}|_{V_ j} shows that r_{\mathcal{L}, \psi }(X_ j) \subset U_ j and that there is a factorization
\xymatrix{ X_ j \ar[r]^-{r_{\mathcal{L}, \psi }} & U_ j \ar[r] & \mathbf{P}(\mathcal{E}_ j) }
We omit the verification. This shows that r_{\mathcal{L}, \psi } is an immersion.
At this point we see that (1), (2) and (4) are equivalent. Clearly (4) implies (3). Assume (3). We will prove (1). Let \mathcal{A} be a quasi-coherent sheaf of graded \mathcal{O}_ S-algebras generated in degree 1 over \mathcal{O}_ S. Consider the map of graded \mathcal{O}_ S-algebras \text{Sym}(\mathcal{A}_1) \to \mathcal{A}. This is surjective by hypothesis and hence induces a closed immersion
\underline{\text{Proj}}_ S(\mathcal{A}) \longrightarrow \mathbf{P}(\mathcal{A}_1)
which pulls back \mathcal{O}(1) to \mathcal{O}(1), see Constructions, Lemma 27.18.5. Hence it is clear that (3) implies (1). \square
Lemma 29.38.8. Let f : X \to S be a morphism of schemes. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. Let S' \to S be a morphism of schemes. Let f' : X' \to S' be the base change of f and denote \mathcal{L}' the pullback of \mathcal{L} to X'. If \mathcal{L} is f-very ample, then \mathcal{L}' is f'-very ample.
Proof. By Definition 29.38.1 there exists there exist a quasi-coherent \mathcal{O}_ S-module \mathcal{E} and an immersion i : X \to \mathbf{P}(\mathcal{E}) over S such that \mathcal{L} \cong i^*\mathcal{O}_{\mathbf{P}(\mathcal{E})}(1). The base change of \mathbf{P}(\mathcal{E}) to S' is the projective bundle associated to the pullback \mathcal{E}' of \mathcal{E} and the pullback of \mathcal{O}_{\mathbf{P}(\mathcal{E})}(1) is \mathcal{O}_{\mathbf{P}(\mathcal{E}')}(1), see Constructions, Lemma 27.16.10. Finally, the base change of an immersion is an immersion (Schemes, Lemma 26.18.2). \square
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Comment #8126 by Jinyong An on
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