Let \(C, D\) be any two divisors on a surface \(X\), and let the corresponding invertible sheaves be \(\mathcal{L}, \mathcal{M}\). Show that \[ C . D=\chi\left(\mathcal{O}_X\right)-\chi\left(\mathcal{L}^{-1}\right)-\chi\left(\mathcal{M}^{-1}\right)+\chi\left(\mathcal{L}^{-1} \otimes \mathcal{M}^{-1}\right) . \]
Solution
For divisors \(A,D\) on \(X\), put \[ \Phi(A,D) = \chi(\OO_X) -\chi(\OO_X(-A)) -\chi(\OO_X(-D)) +\chi(\OO_X(-A-D)). \] The required identity is \[ \Phi(C,D)=C\cdot D. \]
1. Let \(E\subseteq X\) be a nonsingular irreducible curve. For arbitrarydivisors \(A,D\) on \(X\),\[ \Phi(A+E,D)-\Phi(A,D)=E\cdot D. \]
Since \(E\) is an effective Cartier divisor on the nonsingular surface \(X\), there are exact sequences \[ 0 \longrightarrow \OO_X(-A-E) \longrightarrow \OO_X(-A) \longrightarrow \OO_E(-A) \longrightarrow0 \] and \[ 0 \longrightarrow \OO_X(-A-D-E) \longrightarrow \OO_X(-A-D) \longrightarrow \OO_E(-A-D) \longrightarrow0. \] Additivity of Euler characteristic gives \[ \begin{aligned} \Phi(A+E,D)-\Phi(A,D) &= \chi(\OO_E(-A)) -\chi(\OO_E(-A-D)). \end{aligned} \]
For a line bundle \(N\) on the nonsingular projective curve \(E\), D-CRVDEGREES gives \[ \chi(E,N)=\chi(E,\OO_E)+\deg_E N. \] Applying this to \[ N=\OO_E(-A) \] and \[ N\tensor\OO_E(-D)=\OO_E(-A-D) \] yields \[ \chi(\OO_E(-A))- \chi(\OO_E(-A-D)) = \deg_E\OO_E(D). \] By the definition of the surface intersection pairing D-SRFINT, \[ \deg_E\OO_E(D)=E\cdot D. \] This proves the claim.
2. Every divisor \(C\) is linearly equivalent to\[ P-Q \]for nonsingular irreducible effective divisors \(P,Q\).
Choose a very ample divisor \(H\) on \(X\). By the ample/very-ample properties of Exercise II.7.5, the sheaf \[ \OO_X(C+(n-1)H) \] is globally generated for all sufficiently large \(n\): this is the defining large-twist property of the ample sheaf \(\OO_X(H)\). Since \(\OO_X(H)\) is very ample, part (d) of that exercise makes \[ \OO_X(C+nH) = \OO_X(H)\tensor\OO_X(C+(n-1)H) \] very ample. Also \(\OO_X((n-1)H)\) is globally generated for every \(n\ge1\), so the same part (d) makes \(\OO_X(nH)\) very ample. Thus, for all sufficiently large \(n\), both divisor classes \[ C+nH \qquad\text{and}\qquad nH \] are very ample.
Choose general members \[ P\in\abs{C+nH}, \qquad Q\in\abs{nH}. \] Each complete linear system is the hyperplane system of its very ample embedding. Bertini’s hyperplane theorem T-BERTINI therefore makes a general member nonsingular; because \(X\) is irreducible of dimension \(2\), the same theorem makes the general hyperplane section connected. A nonsingular curve is regular, so its distinct irreducible components are disjoint and open-and-closed. Connectedness therefore forces only one component. Thus \(P\) and \(Q\) are nonsingular irreducible effective divisors, and \[ P-Q\sim C. \]
3. For the divisor \(Q\) of step 2,\[ \Phi(-Q,D)=-Q\cdot D. \]
Apply step 1 with \[ A=-Q, \qquad E=Q. \] It gives \[ \Phi(0,D)-\Phi(-Q,D)=Q\cdot D. \] Directly from the definition, \[ \Phi(0,D)=0. \] Hence \[ \Phi(-Q,D)=-Q\cdot D. \]
4. For arbitrary divisors \(C,D\),\[ \Phi(C,D)=C\cdot D. \]
Choose \(P,Q\) as in step 2, so \[ C\sim P-Q. \] The expression \(\Phi(C,D)\) depends only on the invertible sheaves associated to the divisor classes, and the intersection pairing depends only on linear equivalence. Thus it is enough to compute with \(P-Q\).
Apply step 1 with \[ A=-Q, \qquad E=P. \] Then \[ \Phi(P-Q,D)-\Phi(-Q,D)=P\cdot D. \] Using step 3, \[ \Phi(P-Q,D) = P\cdot D-Q\cdot D = (P-Q)\cdot D = C\cdot D. \]
5. Q.E.D.
For the line bundles \[ \mathcal L=\OO_X(C), \qquad \mathcal M=\OO_X(D), \] step 4 reads \[ C\cdot D = \chi(\OO_X) -\chi(\mathcal L^{-1}) -\chi(\mathcal M^{-1}) +\chi(\mathcal L^{-1}\tensor\mathcal M^{-1}), \] which is exactly the desired formula.