Establish the following properties of ample and very ample invertible sheaves on a noetherian scheme \(X\). Here \(\mcl, \mcm\) denote invertible sheaves, and for (d), (e) we assume furthermore that \(X\) is of finite type over a noetherian ring \(A\).
If \(\mcl\) is ample and \(\mcm\) is generated by global sections, then \(\mcl \tensor \mcm\) is ample.
If \(\mcl\) is ample and \(\mcm\) is arbitrary, then \(\mcm \tensor \mcl^n\) is ample for sufficiently large \(n\).
If \(\mcl, \mcm\) are both ample, so is \(\mcl \tensor \mcm\).
If \(\mcl\) is very ample and \(\mcm\) is generated by global sections, then \(\mcl \tensor \mcm\) is very ample.
If \(\mcl\) is ample, then there is an \(n_0 > 0\) such that \(\mcl^n\) is very ample for all \(n \geq n_0\).
Solution
All tensor products of sheaves on \(X\) are over \(\OO_X\), and a power of an invertible sheaf means its tensor power. Use the noetherian definition of ampleness: \(\mathcal L\) is ample when, for every coherent \(\mathcal F\), the sheaf \(\mathcal F\otimes\mathcal L^q\) is globally generated for all sufficiently large integers \(q\). In (d) and (e), very ampleness is relative to \(\Spec A\). The assertions hold on the empty scheme, so assume \(X\ne\varnothing\).
1. Tensor products of globally generated sheaves are globally generated, and every positive power of an ample invertible sheaf is ample.
If global sections \(s_i\) generate \(\mathcal F\) and \(t_j\) generate \(\mathcal G\), the global sections \(s_i\otimes t_j\) generate \(\mathcal F\otimes\mathcal G\). At each stalk this follows by expanding the two factors of an elementary tensor in their respective generators; elementary tensors span the tensor product. In particular, all nonnegative powers of a globally generated invertible sheaf are globally generated, with its zeroth power generated by \(1\).
If \(\mathcal L\) is ample and \(a>0\), then for any coherent \(\mathcal F\) the sheaves \[ \mathcal F\otimes(\mathcal L^a)^q=\mathcal F\otimes\mathcal L^{aq} \] are globally generated for all sufficiently large \(q\), since \(aq\) eventually exceeds the bound in the ample definition for \(\mathcal F\). Thus \(\mathcal L^a\) is ample.
2. Part (a) holds.
1.Their tensor product\[ \mathcal F\otimes(\mcl\otimes\mcm)^q \cong(\mathcal F\otimes\mcl^q)\otimes\mcm^q \]is therefore globally generated for all those \(q\).This is the definition of ampleness of \(\mcl\otimes\mcm\).
3. Part (b) holds for every \(n\ge a+1\), where \(a\ge0\) is any integer for which \(\mcm\otimes\mcl^a\) is globally generated.
1.Apply part (a), proved in step 2, to this sheaf and the globally generated sheaf \(\mcm\otimes\mcl^a\).Their product is \(\mcm\otimes\mcl^n\), which is therefore ample for every such \(n\).
4. Part (c) holds.
For any coherent \(\mathcal F\), ampleness gives an integer beyond which \(\mathcal F\otimes\mcl^q\) is globally generated, and an integer beyond which \(\mcm^q\) is globally generated, using \(\OO_X\) in the second application. Choose \(q\) at least both bounds. Step 1 then makes \[ (\mathcal F\otimes\mcl^q)\otimes\mcm^q \cong\mathcal F\otimes(\mcl\otimes\mcm)^q \] globally generated. The same holds for every larger \(q\), proving that \(\mcl\otimes\mcm\) is ample.
5. In part (d), the graph of the morphism defined by \(\mcm\), followed by a Segre immersion, realizes \(\mcl\otimes\mcm\) as very ample over \(\Spec A\).
Choose an \(A\)-immersion \(i:X\to\PP_A^r\) with \(i^*\OO(1)\cong\mcl\). Because \(\mcm\) is globally generated, the open sets where one global section generates it cover \(X\). Quasi-compactness supplies a finite subcover, so finitely many global sections \(t_0,\ldots,t_s\) generate \(\mcm\). They define an \(A\)-morphism \(g:X\to\PP_A^s\) with \(g^*\OO(1)\cong\mcm\) [@Har10a, Theorem II.7.1].
The morphism \((i,g):X\to\PP_A^r\times_A\PP_A^s\) factors as \[ X\xrightarrow{\Gamma_g}X\times_A\PP_A^s \xrightarrow{i\times\id}\PP_A^r\times_A\PP_A^s. \] The graph \(\Gamma_g\) is a closed immersion because \(\PP_A^s\) is separated over \(A\); it is a base change of its diagonal. The second arrow is a base change of the immersion \(i\) and is an immersion. Their composite is therefore an immersion.
Compose with the Segre closed immersion \[ \sigma:\PP_A^r\times_A\PP_A^s\longrightarrow\PP_A^{(r+1)(s+1)-1}. \] The Segre coordinates are the pairwise products of the coordinates of the two factors, so \[ \sigma^*\OO(1)\cong\operatorname{pr}_1^*\OO(1)\otimes\operatorname{pr}_2^*\OO(1) \] [@Har10a, Exercise II.5.11]. The resulting immersion \(h=\sigma\circ(i,g)\) satisfies \[ h^*\OO(1)\cong i^*\OO(1)\otimes g^*\OO(1)\cong\mcl\otimes\mcm. \] This proves very ampleness over the specified base, without assuming that \(X\) is proper.
6. Part (e) holds for \(n_0=h+b\), where \(h>0\) has \(\mcl^h\) very ample and \(b\ge0\) has \(\mcl^q\) globally generated for every \(q\ge b\).
The integer \(h\) exists by [@Har10a, Theorem II.7.6], since \(X\) is of finite type over the noetherian ring \(A\). The integer \(b\) exists by applying the ample definition to \(\OO_X\). For every \(n\ge h+b\), the factor \(\mcl^{n-h}\) is globally generated. Part (d), proved in step 5, shows that \[ \mcl^n\cong\mcl^h\otimes\mcl^{n-h} \] is very ample. This treats every sufficiently large exponent, not only multiples of \(h\), and \(n_0=h+b\) is positive.
7. Q.E.D.
Steps <1>2–<1>6 prove parts (a)–(e), respectively.