2_7x Projective Morphisms

II.7: Projective Morphisms

II.7.1. #to_work

Let \(\left(X, {\mathcal{O}}_X\right)\) be a locally ringed space, and let \(f: \mathcal{L} \rightarrow \mathcal{M}\) be a surjective map of invertible sheaves on \(X\). Show that \(f\) is an isomorphism. ¹

II.7.2. #to_work

Let \(X\) be a scheme over a field \(k\). Let \(\mathcal{L}\) be an invertible sheaf on \(X\), and let \(\left\{s_0, \ldots, s_n\right\}\) and \(\left\{t_0, \ldots, t_m\right\}\) be two sets of sections of \(\mathcal{L}\), which generate the same subspace \(V \subseteq \Gamma(X, \mathcal{L})\), and which generate the sheaf \(\mathcal{L}\) at every point. Suppose \(n \leqslant m\).

Show that the corresponding morphisms \(\varphi: X \rightarrow \mathbf{P}_k^n\) and \(\psi: X \rightarrow\) \(\mathbf{P}_k^m\) differ by a suitable linear projection \(\mathbf{P}^m-L \rightarrow \mathbf{P}^n\) and an automorphism of \(\mathbf{P}^n\), where \(L\) is a linear subspace of \(\mathbf{P}^m\) of dimension \(m-n-1\).

II.7.3. #to_work

Let \(\varphi: \mathbf{P}_k^n \rightarrow \mathbf{P}_k^m\) be a morphism. Then:

• Either \(\varphi\left(\mathbf{P}^n\right) = {\operatorname{pt}}\) or \(m \geqslant n\) and \(\operatorname{dim} \varphi\left(\mathbf{P}^n\right)=n\);

• In the second case, \(\varphi\) can be obtained as the composition of

  • a \(d\)-uple embedding \(\mathbf{P}^n \rightarrow \mathbf{P}^N\) for a uniquely determined \(d \geqslant 1,\)
  • a linear projection \(\mathbf{P}^N-\mathbf{L} \rightarrow \mathbf{P}^m\), and
  • an automorphism of \(\mathbf{P}^m\).

  Also, \(\varphi\) has finite fibres.

II.7.4. #to_work

• Use (7.6) to show that if \(X\) is a scheme of finite type over a noetherian \(\operatorname{ring} A\), and if \(X\) admits an ample invertible sheaf, then \(X\) is separated.

• Let \(X\) be the affine line over a field \(k\) with the origin doubled (4.0.1). Calculate \(\operatorname{Pic}X\), determine which invertible sheaves are generated by global sections, and then show directly (without using (a)) that there is no ample invertible sheaf on \(X\).

II.7.5. #to_work

Establish the following properties of ample and very ample invertible sheaves on a noetherian scheme \(X\). \(\mathcal{L}, \mathcal{M}\) will denote invertible sheaves, and for (d), (e) we assume furthermore that \(X\) is of finite type over a noetherian ring \(A\).

• If \(\mathcal{L}\) is ample and \(\mathcal{M}\) is generated by global sections, then \(\mathcal{L} \otimes \mathcal{M}\) is ample.

• If \(\mathcal{L}\) is ample and \(\mathcal{M}\) is arbitrary, then \(\mathcal{M} \otimes \mathcal{L}^n\) is ample for sufficiently large \(n\).

• If \(\mathcal{L}, \mathcal{M}\) are both ample, so is \(\mathcal{L} \otimes \mathcal{M}\).

• If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is generated by global sections, then \(\mathcal{L} \otimes \mathcal{M}\) is very ample.

• If \(\mathcal{L}\) is ample, then there is an \(n_0>0\) such that \(\mathcal{L}^n\) is very ample for all \(n \geqslant n_0\).

II.7.6. The Riemann-Roch Problem. #to_work

Let \(X\) be a nonsingular projective variety over an algebraically closed field, and let \(D\) be a divisor on \(X\). For any \(n>0\) we consider the complete linear system \(|n D|\). Then the Riemann-Roch problem is to determine \(\operatorname{dim}|n D|\) as a function of \(n\), and, in particular, its behavior for large \(n\).

If \(\mathcal{L}\) is the corresponding invertible sheaf, then \(\operatorname{dim}|n D|=\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)-1\), so an equivalent problem is to determine \(\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)\) as a function of \(n\).

• Show that if \(D\) is very ample, and if \(X \hookrightarrow \mathbf{P}_k^n\) is the corresponding embedding in projective space, then for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_X(n)-1\), where \(P_X\) is the Hilbert polynomial of \(X(\mathbf{I}, \S 7)\). Thus in this case \(\operatorname{dim}|n D|\) is a polynomial function of \(n\), for \(n\) large.

• If \(D\) corresponds to a torsion element of \(\operatorname{Pic}X\), of order \(r\), then \(\operatorname{dim}|n D|=0\) if \(r \mathrel{\Big|}n\) and \(-1\) otherwise. In this case the function is periodic of period \(r\). It follows from the general Riemann-Roch theorem that \(\operatorname{dim}|n D|\) is a polynomial function for \(n\) large, whenever \(D\) is an ample divisor. ²

  In the case of algebraic surfaces, Zariski \([7]\) has shown for any effective divisor \(D\), that there is a finite set of polynomials \(P_1, \ldots, P_r\), such that for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_{i(n)}(n)\), where \(i(n) \in\{1,2, \ldots, r\}\) is a function of \(n\).

II.7.7. Some Rational Surfaces. #to_work

Let \(X=\mathbf{P}_k^2\), and let \(|D|\) be the complete linear system of all divisors of degree 2 on \(X\) (conics). \(D\) corresponds to the invertible sheaf \({\mathcal{O}}(2)\), whose space of global sections has a basis \(x^2, y^2, z^2, x y, x z, y z\), where \(x, y, z\) are the homogeneous coordinates of \(X\).

• The complete linear system \(|D|\) gives an embedding of \(\mathbf{P}^2\) in \(\mathbf{P}^5\), whose image is the Veronese surface. ³

• Show that the subsystem defined by \(x^2, y^2, z^2, y(x-z),(x-y) z\) gives a closed immersion of \(X\) into \(\mathbf{P}^4\). The image is called the Veronese surface in \(\mathbf{P}^4\). Cf. (IV, Ex. 3.11).

• Let \(\nu \subseteq|D|\) be the linear system of all conics passing through a fixed point \(P\). Then \(\nu\) gives an immersion of \(U=X-P\) into \(\mathbf{P}^4\). Furthermore, if we blow up \(P\), to get a surface \(\tilde{X}\), then this map extends to give a closed immersion of \(\tilde{X}\) in \(\mathbf{P}^4\).

  Show that \(\tilde{X}\) is a surface of degree 3 in \(\mathbf{P}^4\), and that the lines in \(X\) through \(P\) are transformed into straight lines in \(\tilde{X}\) which do not meet. \(\tilde{X}\) is the union of all these lines, so we say \(\tilde{X}\) is a ruled surface \((\mathrm{V}, 2.19 .1)\).

II.7.8. #to_work

Let \(X\) be a noetherian scheme, let \(\mathcal{E}\) be a coherent locally free sheaf on \(X\), and let \(\pi: \mathbf{P}(\mathcal{E}) \rightarrow X\) be the corresponding projective space bundle. Show that there is a natural \(1-1\) correspondence between sections of \(\pi\) (i.e., morphisms \(\sigma: X \rightarrow\) \(\mathbf{P}(\mathcal{E})\) such that \(\left.\pi \circ \sigma=\mathrm{id}_X\right)\) and quotient invertible sheaves \(\mathcal{E} \rightarrow \mathcal{L} \rightarrow 0\) of \(\mathcal{E}\).

II.7.9. #to_work

Let \(X\) be a regular noetherian scheme, and \(\mathcal{E}\) a locally free coherent sheaf of rank \(\geqslant 2\) on \(X\).

• Show that \(\operatorname{Pic}\mathbf{P}(\mathcal{E}) \cong \operatorname{Pic}X \times \mathbf{Z}\).

• If \(\mathcal{E}^{\prime}\) is another locally free coherent sheaf on \(X\), show that \(\mathbf{P}(\mathcal{E}) \cong \mathbf{P}\left(\mathcal{E}^{\prime}\right)\) (over \(X\) ) if and only if there is an invertible sheaf \(\mathcal{L}\) on \(X\) such that \(\mathcal{E}^{\prime} \cong \mathcal{E} \otimes \mathcal{L}\).

II.7.10. \(\mathbf{P}^n\)-Bundles Over a Scheme. #to_work

Let \(X\) be a noetherian scheme.

• By analogy with the definition of a vector bundle (Ex. 5.18), define the notion of a projective \(n\)-space bundle over \(X\), as a scheme \(P\) with a morphism \(\pi: P \rightarrow X\) such that \(P\) is locally isomorphic to \(U \times \mathbf{P}^n, U \subseteq X\) open, and the transition automorphisms on \(\operatorname{Spec} A \times \mathbf{P}^n\) are given by \(A\)-linear automorphisms of the homogeneous coordinate ring \(A\left[x_0, \ldots, x_n\right]\)

  E.g., \(x_i^{\prime}=\sum a_{i j} x_j, a_{i j} \in A\).

• If \(\mathcal{E}\) is a locally free sheaf of rankof rank \(n+1\) on \(X\) then \(\mathbf{P}(\mathcal E)\) is a \(\mathbf P^n{\hbox{-}}\)bundle over \(X\).

• * Assume that \(X\) is regular, and show that every \(\mathbf{P}^n\)-bundle \(P\) over \(X\) is isomorphic to \(\mathbf{P}(\mathcal{E})\) for some locally free sheaf \({\mathcal{E}}\) on \(X\). ⁴

• Conclude (in the case \(X\) regular) that we have a 1-1 correspondence between \(\mathbf{P}^n\)-bundles over \(X\), and equivalence classes of locally free sheaves \(\mathcal{E}\) of rank \(n+1\) under the equivalence relation \({\mathcal{E}}\sim {\mathcal{E}}'\) if and only if \({\mathcal{E}}' \cong {\mathcal{E}}\otimes{\mathcal{M}}\) for some invertible sheaf \({\mathcal{M}}\) on \(X\).

II.7.11. #to_work

On a noetherian scheme \(X\), different sheaves of ideals can give rise to isomorphic blown up schemes.

• If \(\mathcal{I}\) is any coherent sheaf of ideals on \(X\), show that blowing up \(\mathcal{I}^d\) for any \(d \geqslant 1\) gives a scheme isomorphic to the blowing up of \(\mathcal{I}\) (cf. Ex. 5.13).

• If \(\mathcal{I}\) is any coherent sheaf of ideals, and if \(\mathcal{J}\) is an invertible sheaf of ideals, then \(\mathcal{I}\) and \(\mathcal{I} \cdot \mathcal{J}\) give isomorphic blowings-up.

• If \(X\) is regular, show that (7.17) can be strengthened as follows. Let \(U \subseteq X\) be the largest open set such that \(f: f^{-1} U \rightarrow U\) is an isomorphism. Then \(\mathcal{I}\) can be chosen such that the corresponding closed subscheme \(Y\) has support equal to \(X-U\)

II.7.12. #to_work

Let \(X\) be a noetherian scheme, and let \(Y, Z\) be two closed subschemes, neither one containing the other. Let \(\tilde{X}\) be obtained by blowing up \(Y \cap Z\) (defined by the ideal sheaf \(\mathcal{I}_Y+\mathcal{I}_Z\) ). Show that the strict transforms \(\tilde{Y}\) and \(\tilde{Z}\) of \(Y\) and \(Z\) in \(\tilde{X}\) do not meet.

II.7.13. * A Complete Nonprojective Variety. #to_work

Let \(k\) be an algebraically closed field of char \(\neq 2\). Let \(C \subseteq \mathbf{P}_k^2\) be the nodal cubic curve \begin{align*} y^2 z=x^3+x^2 z .\end{align*} If \(P_0=(0,0,1)\) is the singular point, then \(C-P_0\) is isomorphic to the multiplicative group \(\mathbf{G}_m=\operatorname{Spec} k\left[t, t^{-1}\right]\) (Ex. 6.7). For each \(a \in k, a \neq 0\), consider the translation of \(\mathbf{G}_m\) given by \(t \mapsto a t\). This induces an automorphism of \(C\) which we denote by \(\varphi_a\). Now consider \(C \times\left(\mathbf{P}^1-\{0\}\right)\) and \(C \times\left(\mathbf{P}^1-\{\infty\}\right)\). We glue their open subsets \(C \times\left(\mathbf{P}^1-\{0, \infty\}\right)\) by the isomorphism \begin{align*} \varphi:\langle P, u\rangle \mapsto\left\langle\varphi_u(P), u\right\rangle,\qquad P \in C, u \in \mathbf{G}_m=\mathbf{P}^1-\{0, \infty\} .\end{align*} Thus we obtain a scheme \(X\), which is our example. The projections to the second factor are compatible with \(\varphi\), so there is a natural morphism \(\pi: X \rightarrow \mathbf{P}^1\).

• Show that \(\pi\) is a proper morphism, and hence that \(X\) is a complete variety over \(k\).

• Use the method of (Ex. 6.9) to show that ⁵ \begin{align*} \operatorname{Pic}\left(C \times \mathbf{A}^1\right) \cong \mathbf{G}_m \times \mathbf{Z}\quad\text{and}\quad \operatorname{Pic}\left(C \times\left(\mathbf{A}^1-\{0\}\right)\right) \cong \mathbf{G}_m \times \mathbf{Z} \times \mathbf{Z} .\end{align*}

• Now show that the restriction map \begin{align*} \operatorname{Pic}\left(C \times \mathbf{A}^1\right) \rightarrow \operatorname{Pic}\left(C \times\left(\mathbf{A}^1-\{0\}\right)\right) \end{align*} is of the form \(\langle t, n\rangle \mapsto\langle t, 0, n\rangle\), and that the automorphism \(\varphi\) of \(C \times\left(\mathbf{A}^1-\{0\}\right)\) induces a map of the form \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its Picard group.

• Conclude that the image of the restriction map \begin{align*} \operatorname{Pic}X \rightarrow \operatorname{Pic}(C \times\{0\}) \end{align*} consists entirely of divisors of degree 0 on \(C\). Hence \(X\) is not projective over \(k\) and \(\pi\) is not a projective morphism.

II.7.14. #to_work

• Give an example of a noetherian scheme \(X\) and a locally free coherent sheaf \(\mathcal{E}\), such that the invertible sheaf \({\mathcal{O}}(1)\) on \(\mathbf{P}(\mathcal{E})\) is not very ample relative to \(X\).

• Let \(f: X \rightarrow Y\) be a morphism of finite type, let \(\mathcal{L}\) be an ample invertible sheaf on \(X\), and let \(\mathcal{S}\) be a sheaf of graded \({\mathcal{O}}_X\)-algebras satisfying (†). Let \(P=\operatorname{Proj} \mathcal{S}\), let \(\pi: P \rightarrow X\) be the projection, and let \({\mathcal{O}}_P(1)\) be the associated invertible sheaf. Show that for all \(n \gg 0\), the sheaf \({\mathcal{O}}_P(1) \otimes \pi^* \mathcal{L}^n\) is very ample on \(P\) relative to \(Y\). ⁶