Use (4.11) to show that if \(D\) is any ample divisor on the cubic surface \(X\), then \(H^1\left(X, \mathcal{O}_X(-D)\right)=0\). This is Kodaira’s vanishing theorem for the cubic surface (III, 7.15).
Solution
Let \[ X\subseteq\PP^3 \] be the nonsingular cubic surface and let \(D\) be ample.
1. One has\[ \boxed{ H^0(X,\OO_X)=k, \qquad H^1(X,\OO_X)=0.} \]
The cubic surface is \[ X\cong\operatorname{Bl}_{P_1,\ldots,P_6}\PP^2 \] for six points in general position FE-SRFCUBIC. Point blowups of a nonsingular surface preserve \[ q=h^1(\OO) \] FE-SRFBLOW. Since \[ H^1(\PP^2,\OO_{\PP^2})=0, \] it follows that \[ H^1(X,\OO_X)=0. \] Also \(X\) is integral and projective over the algebraically closed ground field, so every global regular function is constant: \[ H^0(X,\OO_X)=k. \]
2. The divisor class \(D\) contains a nonsingular irreducible curve\[ \boxed{C\in|D|.} \]
The ampleness criterion (4.11) gives \[ D\cdot L>0 \] for every one of the \(27\) lines on \(X\), and \[ D^2>0. \] Thus \(D\) satisfies alternative (c) of V.4.8. By P-AGH548IRREDCLASSES, every such class contains an irreducible nonsingular curve. Choose one and call it \(C\). Then \[ C\sim D. \]
3. The line bundle\[ \OO_X(-D) \]has no nonzero global section:\[ \boxed{H^0(X,\OO_X(-D))=0.} \]
2. Thus its zero divisor is a nonzero effective divisor\[ E\sim-D. \]Because \(D\) is ample, it has positive intersection with every nonzeroeffective curve, hence\[ D\cdot E>0. \]But numerical equivalence gives\[ D\cdot E = -D^2 <0, \]since \(D^2>0\) by step
2. This contradiction proves the vanishing.
4. The curve \(C\) gives an exact sequence\[ 0 \longrightarrow \OO_X(-D) \longrightarrow \OO_X \longrightarrow \OO_C \longrightarrow 0. \]
Since \(C\) is an effective Cartier divisor linearly equivalent to \(D\), its ideal sheaf is \[ \mathcal I_C\cong\OO_X(-C)\cong\OO_X(-D). \] The standard ideal-sheaf sequence of \(C\) is therefore exactly the displayed sequence.
5. One has\[ \boxed{H^0(C,\OO_C)=k,} \]and the restriction map\[ H^0(X,\OO_X)\longrightarrow H^0(C,\OO_C) \]is an isomorphism.
The curve \(C\) is nonsingular and irreducible by step 2, hence integral and projective. Thus every global regular function on \(C\) is constant: \[ H^0(C,\OO_C)=k. \] The restriction map sends the constant function \(1\) on \(X\) to the constant function \(1\) on \(C\). Under the identifications \[ H^0(X,\OO_X)=k=H^0(C,\OO_C), \] it is therefore the identity map on \(k\).
6. Consequently\[ \boxed{H^1(X,\OO_X(-D))=0.} \]
The long exact cohomology sequence of step 4 begins \[ 0 \longrightarrow H^0(X,\OO_X(-D)) \longrightarrow H^0(X,\OO_X) \longrightarrow H^0(C,\OO_C) \longrightarrow H^1(X,\OO_X(-D)) \longrightarrow H^1(X,\OO_X). \] By steps 1, 3, and 5 this becomes \[ 0 \longrightarrow 0 \longrightarrow k \xrightarrow{\sim} k \longrightarrow H^1(X,\OO_X(-D)) \longrightarrow 0. \] Exactness forces \[ H^1(X,\OO_X(-D))=0. \]
7. Q.E.D.
Steps <1>1–<1>5 establish the cohomological inputs for the divisor sequence, and step 6 gives the required vanishing.