## 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 \[ \chi(\mathcal{F})=\sum(-1)^i \operatorname{dim}_k H^i(X, \mathcal{F}) . \] If \[ 0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0 \] 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 a. 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 $\mathcal{O}_X(1)$.[^hint.3.5.2] b. 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 $(I, \S 7 )$. [^hint.3.5.2]: Hints: Use induction on dim Supp $\mathcal{F}$, general properties of numerical polynomials (I, 7.3), and suitable exact sequences \[ 0 \rightarrow \mathcal{R} \rightarrow \mathcal{F}(-1) \rightarrow \mathcal{F} \rightarrow \mcl \rightarrow 0 \] ### 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 \[ p_a(X)=(-1)^r\left(\chi\left(\mathcal{O}_X\right)-1\right) . \] Note that it depends only on $X$, not on any projective embedding. a. If $X$ is integral, and $k$ algebraically closed, show that $H^0\left(X, \mathcal{O}_X\right) \cong k$, so that \[ p_a(X)=\sum_{i=0}^{r-1}(-1)^i \operatorname{dim}_k H^{r-i}\left(X, \mathcal{O}_X\right) . \] In particular, if $X$ is a curve, we have[^hint.3.5.3.a] \[ p_a(X)=\operatorname{dim}_k H^1\left(X, \mathcal{O}_X\right) . \] b. 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. c. 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.[^hint.3.5.3.c] [^hint.3.5.3.c]: This gives another proof of (II, 8.20.3) where we used the geometric genus. [^hint.3.5.3.a]: Hint: Use (I, 3.4). ### III.5.4. #to-work Recall from (II, Ex. 6.10) the definition of the Grothendieck group $K(X)$ of a noetherian scheme $X$. a. 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 \[ P: K(X) \rightarrow \mathbf{Q}[z] \] such that for each coherent sheaf $\mathcal{F}$ on $X, P(\gamma(\mathcal{F}))$ is the Hilbert polynomial of $\mathcal{F}$ (Ex. 5.2). b. 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 (1) $K(X)$ is the free abelian group generated by $\left\{\gamma\left(\mathcal{O}_{L_i}\right) \mid i=0, \ldots, r\right\}$, and (2) the map $P: K(X) \rightarrow \mathbf{Q}[z]$ is injective.[^hint.5.4.b.2] [^hint.5.4.b.2]: Hint: Show that (1) $\Rightarrow$ (2). Then prove (1) and (2) simultaneously, by induction on $r$, using (II, Ex. 6.10c). ### 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: a. for all $n \in \mathbf{Z}$, the natural map \[ H^0\left(X, \mathcal{O}_X(n)\right) \rightarrow H^0\left(Y, \mathcal{O}_Y(n)\right) \] is surjective.[^3.5.5.a.hint] b. $Y$ is connected; c. $H^i\left(Y, \mathcal{O}_Y(n)\right)=0$ for $00$, when $Y$ is a disjoint union of $q$ lines $\mathbf{P}^1$ in $Q$, to show: (1) if $|a-b| \leqslant 1$, then $H^1\left(Q, \mathcal{O}_Q(a, b)\right)=0$; (2) if $a, b<0$, then $H^1\left(Q \mathcal{O}_Q(a, b)\right)=0$ (3) If $a \leqslant-2$, then $H^1\left(Q, \mathcal{O}_Q(a, 0)\right) \neq 0$. b. Now use these results to show: (1) if $Y$ is a locally principal closed subscheme of type $(a, b)$, with $a, b>0$, then $Y$ is connected; (2) 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). (3) 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). c. If $Y$ is a locally principal subscheme of type $(a, b)$ in $Q$, show that[^hint.5.6.c] \[ p_a(Y)= a b-a-b+1 .\] [^hint.5.6.c]: Hint: Calculate Hilbert polynomials of suitable sheaves, and again use the special case $(q, 0)$ which is a disjoint union of $q$ copies of $\mathbf{P}^1$. See $(\mathrm{V}, 1.5 .2)$ for another method. ### 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. a. 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. b. $\mathcal{L}$ is ample on $X$ if and only if $\mathcal{L}_{\text {red }}=\mathcal{L} \otimes \mathcal{O}_{X_{\text {red }}}$ is ample on $X_{\text {red }}$. c. 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 $X$. d. 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 $X$.[^hint.3.5.7.d] [^hint.3.5.7.d]: Hints: Use (5.3) and compare (Ex. 3.1, Ex. 3.2, Ex. 4.1, Ex. 4.2). See also Hartshorne $[5, Ch. I \S 4]$ for more details. ### III.5.8. #to-work Prove that every one-dimensional proper scheme $X$ over an algebraically closed field $k$ is projective. a. If $X$ is irreducible and nonsingular, then $X$ is projective by (II, 6.7). b. 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. c. 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 $\Pic X \rightarrow \bigoplus \operatorname{Pic} X_i$ is surjective. Then use (Ex. 5.7c) to show $X$ is projective. d. Finally, if $X$ is any one-dimensional proper scheme over $k$, use (2.7) and (Ex. 4.6) to show that $\Pic X \rightarrow \Pic X_{\text {red }}$ 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 \[ \ldots \rightarrow H^1(X, \omega) \rightarrow \operatorname{Pic} X^{\prime} \rightarrow \operatorname{Pic} X \stackrel{\delta}{\rightarrow} H^2(X, \omega) \rightarrow \ldots \] 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 $\Pic X^{\prime}=0$. In particular, $X^{\prime}$ has no ample invertible sheaves, so it is not projective.[^hint_generalized_3.5.10] [^hint_generalized_3.5.10]: Note. In fact, this result can be generalized to show that for any nonsingular projective surface $X$ over an algebraically closed field $k$ of characteristic 0 , there is an infinitesimal extension $X^{\prime}$ of $X$ by $\omega$, such that $X^{\prime}$ is not projective over $k$. \ Indeed, let $D$ be an ample divisor on $X$. Then $D$ determines an element $c_1(D) \in$ $H^1\left(X, \Omega^1\right)$ which we use to define $X^{\prime}$, as above. Then for any divisor $E$ on $X$ one can show that $\delta(\mathcal{L}(E))=(D . E)$, where $(D . E)$ is the intersection number (Chapter V), considered as an element of $k$. Hence if $E$ is ample, $\delta(\mathcal{L}(E)) \neq 0$. Therefore $X^{\prime}$ has no ample divisors. \ On the other hand, over a field of characteristic $p>0$, a proper scheme $X$ is projective if and only if $X_{\text {red }}$ is! ### 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 \[ \Gamma\left(X, \mathcal{F}^1(n)\right) \rightarrow \Gamma\left(X, \mathcal{F}^2(n)\right) \rightarrow \ldots \rightarrow \Gamma\left(X, \mathcal{F}^r(n)\right) \] is exact.