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III.5: The Cohomology of Projective Space

III.5.1. #to_work

Let \(X\) be a projective scheme over a field \(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). We define the Euler characteristic of \(\mathcal{F}\) by \begin{align*} \chi(\mathcal{F})=\sum(-1)^i \operatorname{dim}_k H^i(X, \mathcal{F}) . \end{align*} If \begin{align*} 0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0 \end{align*} is a short exact sequence of coherent sheaves on \(X\), show that \(\chi(\mathcal{F})=\chi\left(\mathcal{F}^{\prime}\right)+\) \(\chi\left(\mathcal{F}^{\prime \prime}\right)\).

III.5.2. #to_work

• Let \(X\) be a projective scheme over a field \(k\), let \(\mathcal{O}_X(1)\) be a very ample invertible sheaf on \(X\) over \(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). Show that there is a polynomial \(P(z) \in \mathbf{Q}[z]\), such that \(\chi(\mathcal{F}(n))=P(n)\) for all \(n \in \mathbf{Z}\). We call \(P\) the Hilbert polynomial of \(\mathcal{F}\) with respect to the sheaf \(\mathcal{O}_X(1)\). ¹

• Now let \(X=\mathbf{P}_k^r\), and let \(M=\Gamma_*(\mathcal{F})\), considered as a graded \(S=k\left[x_0, \ldots, x_r\right]-\) module. Use (5.2) to show that the Hilbert polynomial of \(\mathcal{F}\) just defined is the same as the Hilbert polynomial of \(M\) defined in \((I, \S 7 )\).

III.5.3. Arithmetic Genus. #to_work

Let \(X\) be a projective scheme of dimension \(r\) over a field \(k\). We define the arithmetic genus \(p_a\) of \(X\) by \begin{align*} p_a(X)=(-1)^r\left(\chi\left(\mathcal{O}_X\right)-1\right) . \end{align*} Note that it depends only on \(X\), not on any projective embedding.

• If \(X\) is integral, and \(k\) algebraically closed, show that \(H^0\left(X, \mathcal{O}_X\right) \cong k\), so that \begin{align*} p_a(X)=\sum_{i=0}^{r-1}(-1)^i \operatorname{dim}_k H^{r-i}\left(X, \mathcal{O}_X\right) . \end{align*} In particular, if \(X\) is a curve, we have ² \begin{align*} p_a(X)=\operatorname{dim}_k H^1\left(X, \mathcal{O}_X\right) . \end{align*}

• If \(X\) is a closed subvariety of \(\mathbf{P}_k^r\), show that this \(p_a(X)\) coincides with the one defined in (I, Ex. 7.2), which apparently depended on the projective embedding.

• If \(X\) is a nonsingular projective curve over an algebraically closed field \(k\), show that \(p_a(X)\) is in fact a birational invariant. Conclude that a nonsingular plane curve of degree \(d \geqslant 3\) is not rational. ³

III.5.4. #to_work

Recall from (II, Ex. 6.10) the definition of the Grothendieck group \(K(X)\) of a noetherian scheme \(X\).

• Let \(X\) be a projective scheme over a field \(k\), and let \(\mathcal{O}_X(1)\) be a very ample invertible sheaf on \(X\). Show that there is a (unique) additive homomorphism \begin{align*} P: K(X) \rightarrow \mathbf{Q}[z] \end{align*} such that for each coherent sheaf \(\mathcal{F}\) on \(X, P(\gamma(\mathcal{F}))\) is the Hilbert polynomial of \(\mathcal{F}\) (Ex. 5.2).

• Now let \(X=\mathbf{P}_k^r\). For each \(i=0,1, \ldots, r\), let \(L_i\) be a linear space of dimension \(i\) in \(X\). Then show that

  • \(K(X)\) is the free abelian group generated by \(\left\{\gamma\left(\mathcal{O}_{L_i}\right) \mathrel{\Big|}i=0, \ldots, r\right\}\), and
  • the map \(P: K(X) \rightarrow \mathbf{Q}[z]\) is injective. ⁴

III.5.5. #to_work

Let \(k\) be a field, let \(X=\mathbf{P}_k^r\), and let \(Y\) be a closed subscheme of dimension \(q \geqslant 1\), which is a complete intersection (II, Ex. 8.4). Then:

• for all \(n \in \mathbf{Z}\), the natural map \begin{align*} H^0\left(X, \mathcal{O}_X(n)\right) \rightarrow H^0\left(Y, \mathcal{O}_Y(n)\right) \end{align*} is surjective. ⁵

• \(Y\) is connected;

• \(H^i\left(Y, \mathcal{O}_Y(n)\right)=0\) for \(0<i<q\) and all \(n \in \mathbf{Z}\);

• \(p_a(Y)=\operatorname{dim}_k H^q\left(Y, \mathcal{O}_Y\right)\). ⁶

III.5.6. Curves on a Nonsingular Quadric Surface. #to_work

Let \(Q\) be the nonsingular quadric surface \(x y=z w\) in \(X=\mathbf{P}_k^3\) over a field \(k\). We will consider locally principal closed subschemes \(Y\) of \(Q\). These correspond to Cartier divisors on \(Q\) by (II, 6.17.1). On the other hand, we know that \(\operatorname{Pic}Q \cong \mathbf{Z} \oplus \mathbf{Z}\), so we can talk about the type \((a, b)\) of \(Y(\) II, 6.16) and (II, 6.6.1).

Let us denote the invertible sheaf \(\mathcal{L}(Y)\) by \(\mathcal{O}_Q(a, b)\). Thus for any \(n \in \mathbf{Z}, \mathcal{O}_Q(n)=\mathcal{O}_Q(n, n)\).

• Use the special cases \((q, 0)\) and \((0, q)\), with \(q>0\), when \(Y\) is a disjoint union of \(q\) lines \(\mathbf{P}^1\) in \(Q\), to show:

  • if \(|a-b| \leqslant 1\), then \(H^1\left(Q, \mathcal{O}_Q(a, b)\right)=0\);
  • if \(a, b<0\), then \(H^1\left(Q \mathcal{O}_Q(a, b)\right)=0\)
  • If \(a \leqslant-2\), then \(H^1\left(Q, \mathcal{O}_Q(a, 0)\right) \neq 0\).

• Now use these results to show:

  • if \(Y\) is a locally principal closed subscheme of type \((a, b)\), with \(a, b>0\), then \(Y\) is connected;
  • now assume \(k\) is algebraically closed. Then for any \(a, b>0\), there exists an irreducible nonsingular curve \(Y\) of type (a,b). Use (II, 7.6.2) and (II, 8.18).
  • an irreducible nonsingular curve \(Y\) of type \((a, b), a, b>0\) on \(Q\) is projectively normal (II, Ex. 5.14) if and only if \(|a-b| \leqslant 1\). In particular, this gives lots of examples of nonsingular, but not projectively normal curves in \(\mathbf{P}^3\). The simplest is the one of type \((1,3)\), which is just the rational quartic curve (I, Ex. 3.18).

• If \(Y\) is a locally principal subscheme of type \((a, b)\) in \(Q\), show that ⁷ \begin{align*} p_a(Y)= a b-a-b+1 .\end{align*}

III.5.7. #to_work

Let \(X\) (respectively, \(Y\) ) be proper schemes over a noetherian ring \(A\). We denote by \(\mathcal{L}\) an invertible sheaf.

• If \(\mathcal{L}\) is ample on \(X\), and \(Y\) is any closed subscheme of \(X\), then \(i^* \mathcal{L}\) is ample on \(Y\), where \(i: Y \rightarrow X\) is the inclusion.

• \(\mathcal{L}\) is ample on \(X\) if and only if \(\mathcal{L}_{\text {red }}=\mathcal{L} \otimes \mathcal{O}_{X_{\text {red }}}\) is ample on \(X_{\text {red }}\).

• Suppose \(X\) is reduced. Then \(\mathcal{L}\) is ample on \(X\) if and only if \(\mathcal{L} \otimes \mathcal{O}_{X_i}\) is ample on \(X_i\), for each irreducible component \(X_i\) of \(X\).

• Let \(f: X \rightarrow Y\) be a finite surjective morphism, and let \(\mathcal{L}\) be an invertible sheaf on \(Y\). Then \(\mathcal{L}\) is ample on \(Y\) if and only if \(f^* \mathcal{L}\) is ample on \(X\). ⁸

III.5.8. #to_work

Prove that every one-dimensional proper scheme \(X\) over an algebraically closed field \(k\) is projective.

• If \(X\) is irreducible and nonsingular, then \(X\) is projective by (II, 6.7).

• If \(X\) is integral, let \(\tilde{X}\) be its normalization (II, Ex. 3.8). Show that \(\tilde{X}\) is complete and nonsingular, hence projective by (a).

  Let \(f: \tilde{X} \rightarrow X\) be the projection. Let \(\mathcal{L}\) be a very ample invertible sheaf on \(\tilde{X}\). Show there is an effective divisor \(D=\sum P_i\) on \(\tilde{X}\) with \(\mathcal{L}(D) \cong \mathcal{L}\), and such that \(f\left(P_i\right)\) is a nonsingular point of \(X\), for each \(i\).

  Conclude that there is an invertible sheaf \(\mathcal{L}_0\) on \(X\) with \(f^* \mathcal{L}_0 \cong\) \(\mathcal{L}\). Then use (Ex. 5.7d), (II, 7.6) and (II, 5.16.1) to show that \(X\) is projective.

• If \(X\) is reduced, but not necessarily irreducible, let \(X_1, \ldots, X_r\) be the irreducible components of \(X\). Use (Ex. 4.5) to show \(\operatorname{Pic}X \rightarrow \bigoplus \operatorname{Pic} X_i\) is surjective. Then use (Ex. 5.7c) to show \(X\) is projective.

• Finally, if \(X\) is any one-dimensional proper scheme over \(k\), use (2.7) and (Ex. 4.6) to show that \(\operatorname{Pic}X \rightarrow \operatorname{Pic}X_{\text {red }}\) is surjective. Then use (Ex. 5.7b) to show \(X\) is projective.

III.5.9. A Nonprojective Scheme. #to_work

We show the result of (Ex. 5.8) is false in dimension 2. Let \(k\) be an algebraically closed field of characteristic 0 , and let \(X=\mathbf{P}_k^2\). Let \(\omega\) be the sheaf of differential 2-forms (II, §8). Define an infinitesimal extension \(X^{\prime}\) of \(X\) by \(\omega\) by giving the element \(\xi \in H^1(X, \omega \otimes \mathcal{T})\) defined as follows (Ex. 4.10).

Let \(x_0, x_1, x_2\) be the homogeneous coordinates of \(X\), let \(U_0, U_1, U_2\) be the standard open covering, and let \(\xi_{i j}=\left(x_j / x_i\right) d\left(x_i / x_j\right)\). This gives a Čech 1-cocycle with values in \(\Omega_X^1\), and since \(\operatorname{dim} X=2\), we have \(\omega \otimes \mathcal{T} \cong \Omega^1\) (II, Ex. 5.16b). Now use the exact sequence \begin{align*} \ldots \rightarrow H^1(X, \omega) \rightarrow \operatorname{Pic} X^{\prime} \rightarrow \operatorname{Pic} X \stackrel{\delta}{\rightarrow} H^2(X, \omega) \rightarrow \ldots \end{align*} of (Ex. 4.6) and show \(\delta\) is injective. We have \(\omega \cong \mathcal{O}_X(-3)\) by (II, 8.20.1), so \(H^2(X, \omega) \cong k\). Since char \(k=0\), you need only show that \(\delta(\mathcal{O}(1)) \neq 0\), which can be done by calculating in Čech cohomology.

Since \(H^1(X, \omega)=0\), we see that \(\operatorname{Pic}X^{\prime}=0\). In particular, \(X^{\prime}\) has no ample invertible sheaves, so it is not projective. ⁹

III.5.10. #to_work

Let \(X\) be a projective scheme over a noetherian ring \(A\), and let \(\mathcal{F}^1 \rightarrow \mathcal{F}^2 \rightarrow \ldots \rightarrow\) \(\mathcal{F}^r\) be an exact sequence of coherent sheaves on \(X\). Show that there is an integer \(n_0\), such that for all \(n \geqslant n_0\), the sequence of global sections \begin{align*} \Gamma\left(X, \mathcal{F}^1(n)\right) \rightarrow \Gamma\left(X, \mathcal{F}^2(n)\right) \rightarrow \ldots \rightarrow \Gamma\left(X, \mathcal{F}^r(n)\right) \end{align*} is exact.