Give an example of a noetherian scheme \(X\) and a locally free coherent sheaf \(\mce\) such that the invertible sheaf \(\OO(1)\) on \(\PP(\mce)\) is not very ample relative to \(X\).
Let \(X\) be noetherian, let \(f:X\to Y\) be a morphism of finite type, and let \(\mcl\) be an ample invertible sheaf on \(X\). Let \(\mcs=\bigoplus_{d\ge0}\mcs_d\) satisfy the relative-Proj condition (†): it is a quasi-coherent graded \(\OO_X\)-algebra with \(\mcs_0=\OO_X\), its degree-one sheaf \(\mcs_1\) is coherent, and the multiplication map \(\operatorname{Sym}^{\bullet}\mcs_1\to\mcs\) is surjective. Let \(P = \Proj \mcs\), let \(\pi: P \to X\) be the projection, and let \(\OO_P(1)\) be the associated invertible sheaf. Show that for all \(n \gg 0\), the sheaf \(\OO_P(1) \tensor \pi^* \mcl^n\) is very ample on \(P\) relative to \(Y\).
Hint
Use Proposition II.7.10 and Exercise II.5.12.
Solution
Very ampleness over a scheme \(B\) means an immersion into \(\PP_B^N\) for some finite \(N\), pulling back \(\OO(1)\) to the given sheaf, as in D-MODAMPLE. We use the source’s open-in-closed immersion convention, as in P-AGH2512VERYAMPLE. Tensor powers are indicated by superscripts, and all sheaf tensor products on \(X\) are over \(\OO_X\).
1. An example for (a) is\[ \boxed{X=\PP_k^1,\qquad\mce=\OO_X(-1).} \]
The scheme \(X\) is noetherian, and \(\mce\) is locally free and coherent of rank one. In the quotient convention, the projection \(\PP(\mce)\to X\) is an isomorphism: on every framed open its symmetric algebra is the polynomial algebra in one degree-one variable, whose Proj is that open. The tautological quotient then identifies \(\OO_{\PP(\mce)}(1)\) with \(\mce\) [@Har10a, Proposition II.7.12], as constructed in P-AGH278SECTIONSPE.
If this sheaf were very ample relative to \(X\), the coordinate sections from a witnessing immersion into some \(\PP_X^N\) would pull back to finitely many global sections generating \(\OO_X(-1)\). But \(\Gamma(\PP_k^1,\OO(-1))=0\) [@Har10a, Proposition II.5.13]. This contradicts generation of a nonzero invertible sheaf and proves (a).
2. Under the hypotheses in (b), some positive power \(\mcl^h\) is very ample relative to \(Y\).
For \(X=\varnothing\), its empty immersion supplies the assertion, so assume \(X\) is nonempty. The affine-basis criterion for ampleness says that the affine opens \(X_s\), with \(s\) a global section of a positive power of \(\mcl\), form a basis of \(X\). Here \(X_s\) is the open set on which \(s\) is a frame. Choose finitely many such opens \(W_i=X_{s_i}\) covering \(X\), with \(s_i\in\Gamma(X,\mcl^{d_i})\), \(d_i>0\), and each \(W_i\) mapped into an affine open \(V_i=\Spec A_i\) of \(Y\). Finiteness of this cover follows from quasi-compactness of \(X\).
Write \(W_i=\Spec B_i\). The restriction \(W_i\to V_i\) is of finite type, so choose finitely many \(A_i\)-algebra generators \(b_{ij}\) of \(B_i\). The localization formula for sections on the quasi-compact, quasi-separated scheme \(X\) gives sections \[ t_{ij}\in\Gamma(X,\mcl^{d_iq_{ij}}),\qquad t_{ij}|_{W_i}=b_{ij}s_i^{q_{ij}} \] for some integers \(q_{ij}\ge0\). Choose a positive common multiple \(h\) of the \(d_i\) large enough that \(h/d_i\ge q_{ij}\) for every pair. The global sections \[ a_i=s_i^{h/d_i},\qquad c_{ij}=t_{ij}s_i^{h/d_i-q_{ij}} \] of \(\mcl^h\) have ratios \(c_{ij}/a_i=b_{ij}\) on \(W_i\). The \(a_i\) have no common zero, so this finite list generates \(\mcl^h\) and defines a \(Y\)-morphism \(i:X\to\PP_Y^N\) with \(i^*\OO(1)\cong\mcl^h\) [@Har10a, Theorem II.7.1].
The inverse image of the target affine chart over \(V_i\) where the coordinate for \(a_i\) is nonzero is exactly \(W_i\). Its coordinate-ring map onto \(B_i\) is surjective, because its coordinate ratios include every \(b_{ij}\). Thus \(i\) is a closed immersion over each of these target opens. Their union contains the image, so \(i\) is a locally closed immersion. It is quasi-compact because every open subset of the noetherian space \(X\) is quasi-compact. The factorization of a quasi-compact immersion makes it an open immersion into a closed subscheme of \(\PP_Y^N\). This proves very ampleness in the stated convention, with no noetherian or affine hypothesis on \(Y\).
3. For each integer \(a\), twisting the graded pieces gives\[ \mcs^{[a]}=\bigoplus_{d\ge0}\mcs_d\otimes\mcl^{ad}, \qquad\operatorname{Proj}_X\mcs^{[a]}\cong P, \]and its twisting sheaf corresponds to\[ N_a\coloneqq\OO_P(1)\otimes\pi^*\mcl^a. \]
On an open set with a frame \(e\) for \(\mcl\), identify the degree-\(d\) piece with \(\mcs_d\) by writing a section as \(s\otimes e^{ad}\). These identifications respect products and give local graded-algebra isomorphisms. Replacing \(e\) by \(ue\) changes degree \(d\) by the factor \(u^{ad}\). The factors cancel in every degree-zero homogeneous fraction, so the induced Proj isomorphisms agree on overlaps.
For the twisting sheaf, a homogeneous local fraction has numerator degree one greater than its denominator degree. Its change of frame retains the factor \(u^a\), exactly the transition factor of \(\pi^*\mcl^a\). Thus the twisting sheaves glue with the displayed extra tensor factor. This proves both assertions, which are the graded twist of [@Har10a, Lemma II.7.9].
4. There is an integer \(a_0\ge0\) such that, for every \(a\ge a_0\), \(N_a\) is the pullback of \(\OO(1)\) under a closed immersion \(j_a:P\to\PP_X^{r_a}\) for some finite \(r_a\).
3.This is the construction underlying [@Har10a, Proposition II.7.10].
5. The conclusion in (b) holds for every \(n\ge a_0+h\).
2.For \(n\ge a_0+h\), put \(a=n-h\) and take the closed immersion \(j_a\) from step
4.Base-changing \(i\) and then applying the Segre closed immersion gives the composite\[ P\xrightarrow{j_a}\PP_X^{r_a} \longrightarrow\PP_Y^{r_a}\times_Y\PP_Y^N \longrightarrow\PP_Y^{(r_a+1)(N+1)-1}. \]It is a locally closed immersion, and the Segre twisting formula from P-AGH2512VERYAMPLE, step 1, shows that its pullback of \(\OO(1)\) is\[ N_a\otimes\pi^*i^*\OO(1) \cong\OO_P(1)\otimes\pi^*\mcl^{a+h} =\OO_P(1)\otimes\pi^*\mcl^n. \]The scheme \(P\) is noetherian: over a finite affine cover of \(X\), it is a closed subscheme of a finite-dimensional projective space, since \(\mcs_1\) is coherent and generates the algebra.The composite is therefore quasi-compact, so the same immersion factorization used in step 2 supplies the source’s open-in-closed convention.It witnesses the required relative very ampleness for every stated \(n\), not just a subsequence of exponents.If \(P\) or \(X\) is empty, the empty immersion gives the assertion directly.
6. Q.E.D.
Step 1 proves (a); steps <1>2–<1>5 construct the required immersions over \(Y\) for all sufficiently large powers, proving (b).