Let \(Y\) be a noetherian scheme, and let \(\mce\) be a locally free \(\mco_Y\dash\)module of rank \(n+1\), with \(n \geq 1\). Let \(X = \PP(\mce)\) (II, §7), with the invertible sheaf \(\mco_X(1)\) and the projection morphism \(\pi: X \to Y\).
Show that:
\(\pi_*(\mco(l)) \cong \operatorname{Sym}^l(\mce)\) for \(l \geq 0\), and \(\pi_*(\mco(l)) = 0\) for \(l < 0\) (II, 7.11);
\(R^i \pi_*(\mco(l)) = 0\) for \(0 < i < n\) and all \(l \in \ZZ\); and
\(R^n \pi_*(\mco(l)) = 0\) for \(l > -n-1\).
Show there is a natural exact sequence \[ 0 \to \Omega_{X/Y} \to (\pi^* \mce)(-1) \to \mco \to 0 ,\] cf. (II, 8.13), and conclude that the relative canonical sheaf \(\omega_{X/Y} = \wedge^n \Omega_{X/Y}\) is isomorphic to \((\pi^* \wedge^{n+1} \mce)(-n-1)\). Show furthermore that there is a natural isomorphism \(R^n \pi_*(\omega_{X/Y}) \cong \mco_Y\) (cf. (7.1.1)).
Now show, for any \(l \in \ZZ\), that \[ R^n \pi_*(\mco(l)) \cong \pi_*\dualof{(\mco(-l-n-1))} \tensor \dualof{(\wedge^{n+1} \mce)} .\]
Assume additionally that \(Y\) is a nonsingular projective variety over an algebraically closed field \(k\). Show that \(p_a(X) = (-1)^n p_a(Y)\) (use (Ex. 8.1)) and \(p_g(X) = 0\) (use (II, 8.11)).
In particular, if \(Y\) is a nonsingular projective curve of genus \(g\), and \(\mce\) a locally free sheaf of rank \(2\), then \(X\) is a projective surface with \(p_a = -g\), \(p_g = 0\), and irregularity \(g\) (7.12.3). This kind of surface is called a geometrically ruled surface (V, §2).
(Scope of the genus assertions)
The source states part (d) after assuming only that \(Y\) is noetherian, but at this point in Hartshorne the invariants \(p_a\) and \(p_g\) used there are invariants of projective varieties, and the formula for \(p_g\) uses nonsingularity. The additional hypothesis stated in part (d) is therefore necessary for the literal assertions made there; parts (a)–(c) retain the source’s original noetherian-base generality.
Solution
Write \[ D=\det\mce=\bigwedge^{n+1}\mce. \] All statements in parts (a)–(c) are local on \(Y\), so we repeatedly use affine open subsets \(U=\Spec A\) on which \(\mce|_U\cong\OO_U^{n+1}\); then \[ \pi^{-1}(U)\cong\PP_A^n. \]
1. For every \(l\in\ZZ\),\[ \pi_*\OO_X(l)\cong \begin{cases} \operatorname{Sym}^l\mce,&l\ge0,\\ 0,&l<0, \end{cases} \]and\[ R^i\pi_*\OO_X(l)=0 \]for \(0<i<n\) and every \(l\), and for \(i=n\) whenever \(l>-n-1\).
The formula for \(\pi_*\OO_X(l)\) is Hartshorne II.7.11 with the convention \[ X=\PP(\mce)=\underline{\operatorname{Proj}}_Y(\operatorname{Sym}\mce). \]
For the higher direct images, Proposition III.8.1 says that \(R^i\pi_*\OO_X(l)\) is associated to the presheaf \[ U\longmapsto H^i(\pi^{-1}(U),\OO_X(l)|_{\pi^{-1}(U)}). \] On an affine trivializing open \(U=\Spec A\), this is \[ H^i(\PP_A^n,\OO_{\PP_A^n}(l)). \] Theorem III.5.1 gives \[ H^i(\PP_A^n,\OO(l))=0 \] for \(0<i<n\) and all \(l\), and also for \(i=n\) when \(l>-n-1\). Since such trivializing affine opens form a basis of \(Y\), the corresponding higher-direct-image sheaves vanish. This proves part (a).
2. There is a natural relative Euler sequence\[ \boxed{ 0\longrightarrow\Omega_{X/Y} \longrightarrow(\pi^*\mce)(-1) \longrightarrow\OO_X \longrightarrow0}. \]
Hartshorne II.7.11 gives the universal quotient \[ \pi^*\mce\twoheadrightarrow\OO_X(1). \] Tensoring by \(\OO_X(-1)\) gives a surjection \[ (\pi^*\mce)(-1)\twoheadrightarrow\OO_X. \] Let its kernel be \(\mck\). On every affine trivializing open \(U\subseteq Y\), the restriction to \[ \pi^{-1}(U)\cong\PP_U^n \] is the usual Euler sequence for relative projective space from II.8.13, whose kernel is \(\Omega_{\PP_U^n/U}\). These local identifications are natural under change of trivialization and therefore glue, giving \[ \mck\cong\Omega_{X/Y}. \]
3. The relative canonical sheaf is\[ \boxed{ \omega_{X/Y}=\bigwedge^n\Omega_{X/Y} \cong(\pi^*D)(-n-1)}. \]
In step 2, the middle term has rank \(n+1\) and the quotient has rank one. Taking determinants in the short exact sequence gives \[ \det\Omega_{X/Y} \cong \det\bigl((\pi^*\mce)(-1)\bigr). \] For a rank-\((n+1)\) bundle, \[ \det\bigl((\pi^*\mce)(-1)\bigr) \cong \pi^*(\det\mce)\otimes\OO_X(-1)^{\otimes(n+1)}. \] Thus \[ \omega_{X/Y}\cong(\pi^*D)(-n-1). \]
4. There is a natural trace isomorphism\[ \boxed{R^n\pi_*\omega_{X/Y}\cong\OO_Y}. \]
On a trivializing affine open \(U=\Spec A\), step 3 identifies \[ \omega_{X/Y}|_{\pi^{-1}(U)} \cong \OO_{\PP_A^n}(-n-1)\otimes_A D(U). \] Theorem III.5.1 gives the canonical top-cohomology trace \[ H^n(\PP_A^n,\OO(-n-1))\cong A, \] represented in the standard Čech cover by the class of \[ (x_0x_1\cdots x_n)^{-1}. \] If the frame of \(\mce\) changes by a matrix \(M\in\operatorname{GL}_{n+1}(A)\), the projective coordinates change by \(M\), and this top Čech class changes by \((\det M)^{-1}\). The local generator of \(D=\det\mce\) changes by \(\det M\). Hence the two factors cancel. The local identifications \[ R^n\pi_*\omega_{X/Y}|_U\cong\OO_U \] therefore agree on overlaps and glue to a natural global isomorphism. This proves the last assertion of part (b).
5. Put \(m=-l-n-1\).Multiplication of twists and the trace of step 4 give a natural pairing\[ \pi_*\OO_X(m)\otimes R^n\pi_*\OO_X(l) \longrightarrow D^\vee. \]
Multiplication of sections gives \[ \OO_X(m)\otimes\OO_X(l)\longrightarrow\OO_X(-n-1). \] Step 3 and the projection formula from Exercise III.8.3 yield \[ R^n\pi_*\omega_{X/Y} \cong R^n\pi_*\OO_X(-n-1)\otimes D. \] Combining with step 4 gives \[ R^n\pi_*\OO_X(-n-1)\cong D^\vee. \] Pushing forward the multiplication map and then using this identification produces the displayed pairing.
6. The pairing of step 5 is perfect, and therefore for every \(l\in\ZZ\),\[ \boxed{ R^n\pi_*\OO_X(l) \cong \bigl(\pi_*\OO_X(-l-n-1)\bigr)^\vee\otimes D^\vee}. \]
The assertion is local on \(Y\). Over a trivializing affine open \(U=\Spec A\), it becomes the standard pairing \[ H^0(\PP_A^n,\OO(m)) \otimes_A H^n(\PP_A^n,\OO(-m-n-1)) \longrightarrow H^n(\PP_A^n,\OO(-n-1))\cong A. \] Theorem III.5.1 describes these modules by monomials and shows that this pairing is perfect: a degree-\(m\) monomial is paired with the unique complementary negative monomial whose product is \((x_0\cdots x_n)^{-1}\).
If \(m<0\), the left \(H^0\) vanishes and step 1 gives the corresponding top-cohomology vanishing, so the statement remains valid. The perfect local pairings are exactly the restrictions of the global pairing in step 5, hence glue to the asserted sheaf isomorphism. This proves part (c).
7. Under the hypotheses of part (d),\[ \boxed{p_a(X)=(-1)^n p_a(Y)}. \]
Step 1 with \(l=0\) gives \[ \pi_*\OO_X=\OO_Y, \qquad R^i\pi_*\OO_X=0\quad(i>0). \] Indeed the intermediate groups vanish by step 1, and the top group vanishes because \(0>-n-1\). Exercise III.8.1 therefore gives natural isomorphisms \[ H^i(X,\OO_X)\cong H^i(Y,\OO_Y) \] for every \(i\). Hence \[ \chi(\OO_X)=\chi(\OO_Y). \]
Let \(m=\dim Y\), so \(\dim X=m+n\). For a projective variety \(V\), \[ p_a(V)=(-1)^{\dim V}\bigl(\chi(\OO_V)-1\bigr). \] Therefore \[ \begin{aligned} p_a(X) &=(-1)^{m+n}(\chi(\OO_X)-1)\\ &=(-1)^n(-1)^m(\chi(\OO_Y)-1)\\ &=(-1)^n p_a(Y). \end{aligned} \]
8. Under the hypotheses of part (d),\[ \boxed{p_g(X)=0}. \]
Since \(Y\) is nonsingular and \(\pi\) is a projective-space bundle, \(X\) is nonsingular. The exact sequence of differentials for the smooth morphism \(\pi\) is \[ 0\longrightarrow\pi^*\Omega_Y \longrightarrow\Omega_X \longrightarrow\Omega_{X/Y} \longrightarrow0. \] Taking determinants gives \[ \omega_X\cong\pi^*\omega_Y\otimes\omega_{X/Y}. \] By step 3, \[ \omega_X \cong \pi^*(\omega_Y\otimes D)\otimes\OO_X(-n-1). \] The ordinary projection formula and step 1 give \[ \pi_*\omega_X \cong (\omega_Y\otimes D)\otimes\pi_*\OO_X(-n-1) =0. \] Thus \[ H^0(X,\omega_X) =H^0(Y,\pi_*\omega_X)=0, \] so \(p_g(X)=0\).
9. If \(Y\) is a nonsingular projective curve of genus \(g\) and \(\operatorname{rk}\mce=2\), then \(X\) is a geometrically ruled surface with\[ \boxed{p_a(X)=-g,\qquad p_g(X)=0,\qquad q(X)=g}. \]
Here \(n=1\), so \(\dim X=2\). The morphism \(\pi:X\to Y\) is projective, and \(Y\) is projective over \(k\), hence \(X\) is projective. It is nonsingular because it is locally \(\PP^1\) over the nonsingular curve \(Y\).
For a nonsingular projective curve, \[ p_a(Y)=g. \] Step 7 therefore gives \[ p_a(X)=-g, \] while step 8 gives \(p_g(X)=0\). For a nonsingular projective surface, Remark III.7.12.3 gives \[ q=p_g-p_a. \] Hence \[ q(X)=0-(-g)=g. \] This is the asserted geometrically ruled surface.
10. Q.E.D.
Step 1 proves part (a), steps <1>2–<1>4 prove part (b), steps <1>5–<1>6 prove part (c), steps <1>7–<1>8 prove part (d), and step 9 proves part (e).