Prove the following theorem of Chern and Griffiths. Let \(X\) be a nonsingular surface of degree \(d\) in \(\PP_{\CC}^{n+1}\), which is not contained in any hyperplane. If \(d<2 n\), then \(p_g(X)=0\). If \(d=2 n\), then either \(p_g(X)=0\), or \(p_g(X)=1\) and \(X\) is a K3 surface.
Hint: Cut \(X\) with a hyperplane and use Clifford’s theorem (IV, 5.4). For the last statement, use the Riemann-Roch theorem on \(X\) and the Kodaira vanishing theorem (III, 7.15).
Solution
Put \[ A=\OO_X(1). \] Choose a general hyperplane section \[ C\in|A|. \] By Bertini's theorem, \(C\) is a smooth irreducible curve. Write \[ L=A|_C=\OO_C(1). \] Then \[ \deg L=C^2=A^2=d. \]
1. The hyperplane line bundle satisfies\[ \boxed{h^0(C,L)\ge n+1.} \]
Since \(X\) is not contained in any hyperplane, restriction of ambient linear forms gives an injection \[ H^0(\PP^{n+1},\OO(1)) \hookrightarrow H^0(X,A). \] Thus \[ h^0(X,A)\ge n+2. \]
The section defining \(C\) gives the exact sequence \[ 0\longrightarrow\OO_X \longrightarrow A \longrightarrow L \longrightarrow0. \] Since \(X\) is an irreducible projective variety, \[ h^0(X,\OO_X)=1. \] The image of \(H^0(X,A)\) in \(H^0(C,L)\) therefore has dimension \[ h^0(X,A)-1\ge n+1. \] Hence \(h^0(C,L)\ge n+1\).
2. If \(d<2n\), then \(L\) is nonspecial:\[ \boxed{H^1(C,L)=0.} \]
1.Thus \(L\) is nonspecial.
3. Whenever \(H^1(C,L)=0\), one has\[ \boxed{p_g(X)=0.} \]
Adjunction gives \[ \omega_C \cong (\omega_X\otimes A)|_C, \] and therefore \[ \omega_X|_C \cong \omega_C\otimes L^{-1}. \] Serre duality on \(C\) gives \[ H^0(C,\omega_C\otimes L^{-1}) \cong H^1(C,L)^\vee =0. \]
For every \(m\ge0\), tensor the hyperplane-section sequence by \(\omega_X\otimes A^{-m}\): \[ 0 \longrightarrow \omega_X\otimes A^{-(m+1)} \longrightarrow \omega_X\otimes A^{-m} \longrightarrow \omega_C\otimes L^{-(m+1)} \longrightarrow0. \] The right-hand sheaf has no nonzero global section. Indeed, if it had a section, multiplying by any nonzero section of \(L^m\) would give a nonzero section of \[ \omega_C\otimes L^{-1}, \] which has none. Hence for every \(m\ge0\), \[ H^0(X,\omega_X\otimes A^{-(m+1)}) \xrightarrow{\sim} H^0(X,\omega_X\otimes A^{-m}). \] Iterating, \[ h^0(X,\omega_X) = h^0(X,\omega_X\otimes A^{-m}) \] for every \(m\).
For \(m\gg0\), the latter space is zero. Otherwise an effective divisor \(D_m\) would satisfy \[ D_m\sim K_X-mA, \] so ampleness of \(A\) would give \[ 0\le A\cdot D_m = A\cdot K_X-mA^2, \] which is impossible for sufficiently large \(m\) because \(A^2=d>0\). Therefore \[ p_g(X)=h^0(X,\omega_X)=0. \]
4. If \(d=2n\) and \(L\) is special, then\[ \boxed{L\cong\omega_C.} \]
Clifford’s theorem and step 1 give \[ n+1 \le h^0(C,L) \le \frac d2+1 = n+1. \] Thus equality holds in Clifford’s theorem.
The divisor class \(L\) is nonzero. If equality came from the hyperelliptic case, then \(L\) would be a multiple of the unique \(g^1_2\): \[ L\cong f^*\OO_{\PP^1}(n) \] for the hyperelliptic double cover \[ f:C\to\PP^1. \] Pullback gives an injective map \[ H^0(\PP^1,\OO_{\PP^1}(n)) \hookrightarrow H^0(C,L). \] Both spaces have dimension \(n+1\): the source has that dimension, and equality in the preceding Clifford estimate gives \[ h^0(C,L)=n+1. \] Thus every section of \(L\) is pulled back from \(\PP^1\) and takes the same value on the two points of each fiber of \(f\). Hence \(L\) could not be very ample. But \(L=\OO_C(1)\) is very ample because it is the restriction of the projective embedding.
The only remaining equality case in Clifford’s theorem is therefore \[ L\cong\omega_C. \]
5. In the special case of step 4,\[ \boxed{K_X\cdot A=0.} \]
Adjunction gives \[ \omega_X|_C \cong \omega_C\otimes L^{-1}. \] Step 4 identifies the right-hand side with \(\OO_C\). Consequently \[ \deg(\omega_X|_C)=0. \] Since \(C\sim A\), this degree is \[ K_X\cdot A. \]
6. If \(d=2n\), \(L\) is special, and \(p_g(X)>0\), then\[ \boxed{\omega_X\cong\OO_X} \]and\[ \boxed{p_g(X)=1.} \]
Choose a nonzero canonical section. Its zero divisor \(D\) is effective and satisfies \[ D\sim K_X. \] By step 5, \[ A\cdot D = A\cdot K_X =0. \] Since \(A\) is ample, every nonzero effective curve has positive intersection with \(A\). Hence \(D=0\).
A nowhere-vanishing section of \(\omega_X\) trivializes it, so \[ \omega_X\cong\OO_X. \] Because \(X\) is an irreducible projective variety over \(\CC\), \[ h^0(X,\OO_X)=1. \] Thus \[ p_g(X) = h^0(X,\omega_X) =1. \]
7. Under the hypotheses of step 6,\[ \boxed{q(X)=h^1(X,\OO_X)=0.} \]
Put \[ q=h^1(X,\OO_X). \] Step 6 gives \(p_g=1\), so \[ \chi(\OO_X) = 1-q+p_g = 2-q. \]
Apply surface Riemann–Roch T-SRFRR to \(A\). Since \(K_X\sim0\) and \[ A^2=\deg X=d=2n, \] one obtains \[ \chi(A) = \chi(\OO_X)+\frac12A\cdot(A-K_X) = (2-q)+n = n+2-q. \]
By Kodaira vanishing T-KODVAN, using \[ A\cong\omega_X\otimes A \] and the ampleness of \(A\), \[ H^i(X,A)=0 \qquad (i>0). \] Hence \[ h^0(X,A) = \chi(A) = n+2-q. \]
The given nondegenerate embedding \[ X\hookrightarrow\PP^{n+1} \] supplies \(n+2\) linearly independent global sections of \(A\). Therefore \[ n+2 \le h^0(X,A) = n+2-q. \] Thus \(q\le0\), and hence \[ q=0. \]
8. If \(d<2n\), then \(p_g(X)=0\). If \(d=2n\), then either\(p_g(X)=0\), or \(p_g(X)=1\) and \(X\) is a K3 surface.
For \(d<2n\), step 2 makes \(L\) nonspecial, and step 3 gives \[ p_g(X)=0. \]
Now assume \(d=2n\). If \(L\) is nonspecial, step 3 again gives \(p_g(X)=0\). If \(L\) is special, step 4 gives \(L\cong\omega_C\). If in addition \(p_g(X)=0\), this is the first alternative in the theorem. Otherwise steps <1>6–<1>7 give \[ \omega_X\cong\OO_X, \qquad p_g(X)=1, \qquad q(X)=0. \] A smooth projective surface over \(\CC\) with trivial canonical bundle and \(H^1(X,\OO_X)=0\) is a K3 surface. Hence the second alternative is exactly that \(X\) is K3.
9. Q.E.D.
Steps <1>1–<1>3 prove the strict inequality case. Steps <1>4–<1>8 analyze the equality case and give precisely the two alternatives stated in the theorem.