V: Surfaces

V.1: Geometry on a Surface

V.1.1. #to_work

Let \(C, D\) be any two divisors on a surface \(X\), and let the corresponding invertible sheaves be \(\mathcal{L}, \mathcal{M}\). Show that \begin{align*} 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) . \end{align*}

V.1.2. #to_work

Let \(H\) be a very ample divisor on the surface \(X\), corresponding to a projective embedding \(X \subseteq \mathbf{P}^N\). If we write the Hilbert polynomial of \(X\) (III, Ex. 5.2) as \begin{align*} P(z)=\frac{1}{2} a z^2+b z+c \end{align*} show that \(a=H^2, b=\frac{1}{2} H^2+1-\pi\), where \(\pi\) is the genus of a nonsingular curve representing \(H\), and \(c=1+p_a\). Thus the degree of \(X\) in \(\mathbf{P}^N\), as defined in (I, §7), is just \(H^2\). Show also that if \(C\) is any curve in \(X\), then the degree of \(C\) in \(\mathbf{P}^N\) is just \(C . H\)

V.1.3. #to_work

Recall that the arithmetic genus of a projective scheme \(D\) of dimension 1 is defined as \begin{align*} p_a=1-\chi\left(\mathcal{O}_D\right) .\end{align*} See III, Ex. 5.3.

  • If \(D\) is an effective divisor on the surface \(X\), use (1.6) to show that \begin{align*} 2 p_a-2= D.(D+K) .\end{align*}

  • \(p_a(D)\) depends only on the linear equivalence class of \(D\) on \(X\).

  • More generally, for any divisor \(D\) on \(X\), we define the virtual arithmetic genus (which is equal to the ordinary arithmetic genus if \(D\) is effective) by the same formula: \(2 p_a-2=D .(D+K)\). Show that for any two divisors \(C, D\) we have \begin{align*} p_a(-D)=D^2-p_a(D)+2 \end{align*} and \begin{align*} p_a(C+D)=p_a(C)+p_a(D)+C . D-1 . \end{align*}

V.1.4. #to_work

  • If a surface \(X\) of degree \(d\) in \(\mathbf{P}^3\) contains a straight line \(C=\mathbf{P}^1\), show that \(C^2=2-d\)

  • Assume char \(k=0\), and show for every \(d \geqslant 1\), there exists a nonsingular surface \(X\) of degree \(d\) in \(\mathbf{P}^3\) containing the line \(x=y=0\).

V.1.5. #to_work

  • If \(X\) is a surface of degree \(d\) in \(\mathbf{P}^3\), then \begin{align*} K^2=d(d-4)^2 .\end{align*}

  • If \(X\) is a product of two nonsingular curves \(C, C^{\prime}\), of genus \(g, g^{\prime}\) respectively, then \begin{align*} K^2=8(g-1)\left(g^{\prime}-1\right) .\end{align*} Cf. (II, Ex. 8.3).

V.1.6. #to_work

  • If \(C\) is a curve of genus \(g\), show that the diagonal \(\Delta \subseteq C \times C\) has self-intersection \(\Delta^2=2-2 g\). (Use the definition of \(\Omega_{C / k}\) in (II, §8).)

  • Let \(l=C \times \mathrm{pt}\) and \(m=\mathrm{pt} \times C\). If \(g \geqslant 1\), show that \(l, m\), and \(\Delta\) are linearly independent in \(\operatorname{Num}(C \times C)\). Thus \(\operatorname{Num}(C \times C)\) has rank \(\geqslant 3\), and in particular, \begin{align*} \operatorname{Pic}(C \times C) \neq p_1^* \operatorname{Pic} C \oplus p_2^* \operatorname{Pic}C .\end{align*} Cf. (III, Ex. 12.6), (V, Ex. 4.10).

V.1.7. Algebraic Equivalence of Divisors. #to_work

Let \(X\) be a surface. Recall that we have defined an algebraic family of effective divisors on \(X\), parametrized by a nonsingular curve \(T\), to be an effective Cartier divisor \(D\) on \(X \times T\), flat over \(T\) (III, 9.8.5). In this case, for any two closed points \(0,1 \in T\), we say the corresponding divisors \(D_0, D_1\) on \(X\) are prealgebraically equivalent.

Two arbitrary divisors are prealgebraically equivalent if they are differences of prealgebraically equivalent effective divisors. Two divisors \(D, D^{\prime}\) are algebraically equivalent if there is a finite sequence \(D=D_0, D_1, \ldots, D_n=D^{\prime}\) with \(D_i\) and \(D_{i+1}\) prealgebraically equivalent for each \(i\).

  • Show that the divisors algebraically equivalent to 0 form a subgroup of Div \(X\).

  • Show that linearly equivalent divisors are algebraically equivalent. 1

  • Show that algebraically equivalent divisors are numerically equivalent. 2 3

V.1.8. Cohomology Class of a Divisor. #to_work

For any divisor \(D\) on the surface \(X\), we define its cohomology class \(c(D) \in H^1\left(X, \Omega_X\right)\) by using the isomorphism Pic \(X \cong\) \(H^1\left(X, \mathcal{O}_X^*\right.\) ) of (III, Ex. 4.5) and the sheaf homomorphism \(d \log : \mathcal{O}^* \rightarrow \Omega_X\) (III, Ex. 7.4c). Thus we obtain a group homomorphism \(c: \operatorname{Pic} X \rightarrow H^1\left(X, \Omega_X\right)\). On the other hand, \(H^1(X, \Omega)\) is dual to itself by Serre duality (III, 7.13), so we have a nondegenerate bilinear map \begin{align*} \langle\quad, \quad\rangle: H^1(X, \Omega) \times H^1(X, \Omega) \rightarrow k . \end{align*}

  • Prove that this is compatible with the intersection pairing, in the following sense: for any two divisors \(D, E\) on \(X\), we have \begin{align*} \langle c(D), c(E)\rangle=(D . E) \cdot 1 \end{align*} in \(k\). 4
  • If char \(k=0\), use the fact that \(H^1\left(X, \Omega_X\right)\) is a finite-dimensional vector space to show that \(\operatorname{Num}X\) is a finitely generated free abelian group.

V.1.9. #to_work

  • If \(H\) is an ample divisor on the surface \(X\), and if \(D\) is any divisor, show that \begin{align*} \left(D^2\right)\left(H^2\right) \leqslant(D . H)^2 \text {. } \end{align*}

  • Now let \(X\) be a product of two curves \(X=C \times C^{\prime}\). Let \(l=C \times \mathrm{pt}\), and \(m=\) pt \(\times C^{\prime}\). For any divisor \(D\) on \(X\), let \(a=D . l, b=D . m\). Then we say \(D\) has type \((a, b)\). If \(D\) has type \((a, b)\), with \(a, b \in \mathbf{Z}\), show that \begin{align*} D^2 \leqslant 2 a b \text {, } \end{align*} and equality holds if and only if \(D \equiv b l+a m\). 5

V.1.10. Weil’s Proof of the Analogue of the Riemann Hypothesis for Curves. #to_work

Let \(C\) be a curve of genus \(g\) defined over the finite field \(\mathbf{F}_q\), and let \(N\) be the number of points of \(C\) rational over \(\mathbf{F}_q\). Then \(N=1-a+q\), with \(|a| \leqslant 2 g \sqrt{q}\).

To prove this, we consider \(C\) as a curve over the algebraic closure \(k\) of \(\mathbf{F}_q\). Let \(f: C \rightarrow C\) be the \(k\)-linear Frobenius morphism obtained by taking \(q\) th powers, which makes sense since \(C\) is defined over \(\mathbf{F}_q\), so \(X_q \cong X\) (See \(V, 2.4 .1)\).

Let \(\Gamma \subseteq C \times C\) be the graph of \(f\), and let \(\Delta \subseteq C \times C\) be the diagonal.

Show that \(\Gamma^2=q(2-2 g)\), and \(\Gamma . \Delta=N\). Then apply (Ex. 1.9) to \(D=r \Gamma+s \Delta\) for all \(r\) and \(s\) to obtain the result. 6

V.1.11. #to_work

In this problem, we assume that \(X\) is a surface for which \(\operatorname{Num}X\) is finitely generated (i.e., any surface, if you accept the Néron-Severi theorem (Ex. 1.7)).

  • If \(H\) is an ample divisor on \(X\), and \(d \in \mathbf{Z}\), show that the set of effective divisors \(D\) with \(D . H=d\), modulo numerical equivalence, is a finite set. 7
  • Now let \(C\) be a curve of genus \(g \geqslant 2\), and use (a) to show that the group of automorphisms of \(C\) is finite, as follows. Given an automorphism \(\sigma\) of \(C\), let \(\Gamma \subseteq X=C \times C\) be its graph. First show that if \(\Gamma \equiv \Delta\), then \(\Gamma=\Delta\), using the fact that \(\Delta^2<0\), since \(g \geqslant 2\) (Ex. 1.6). Then use (a). Cf. (V, Ex. 2.5).

V.1.12. #to_work

If \(D\) is an ample divisor on the surface \(X\), and \(D^{\prime} \equiv D\), then \(D^{\prime}\) is also ample. Give an example to show, however, that if \(D\) is very ample, \(D^{\prime}\) need not be very ample.

Footnotes
1.
Hint: If \((f)\) is a principal divisor on \(X\), consider the principal divisor \((t f-u)\) on \(X \times \mathbf{P}^1\), where \(t, u\) are the homogeneous coordinates on \(\mathbf{P}^1\).
2.
Hint: Use (III, 9.9) to show that for any very ample \(H\), if \(D\) and \(D^{\prime}\) are algebraically equivalent, then \(D . H=D^{\prime} . H\).
3.
Note. The theorem of Néron and Severi states that the group of divisors modulo algebraic equivalence, called the Néron-Severi group, is a finitely generated abelian group. Over \(\mathbf{C}\) this can be proved easily by transcendental methods (App. B, \(\S 5\) ) or as in (Ex. 1.8) below. Over a field of arbitrary characteristic, see Lang and Néron [1] for a proof, and Hartshorne [6] for further discussion. Since \(\operatorname{Num}X\) is a quotient of the Néron-Severi group, it is also finitely generated, and hence free, since it is torsion-free by construction.
4.
Hint: Reduce to the case where \(D\) and \(E\) are nonsingular curves meeting transversally. Then consider the analogous map \(c: \operatorname{Pic} D \rightarrow H^1\left(D, \Omega_D\right)\), and the fact (III, Ex. 7.4) that \(c\) (point) goes to 1 under the natural isomorphism of \(H^1\left(D, \Omega_D\right)\) with \(k\).
5.
Hint: Show that \(H=l+m\) is ample, let \(E=l-m\), let \(D^{\prime}=\left(H^2\right)\left(E^2\right) D-\left(E^2\right)(D \cdot H) H-\left(H^2\right)(D \cdot E) E\), and apply (1.9). This inequality is due to Castelnuovo and Severi. See Grothendieck \([2]\).
6.
See (App. C, Ex. 5.7) for another interpretation of this result.
7.
Hint: Use the adjunction formula, the fact that \(p_a\) of an irreducible curve is \(\geqslant 0\), and the fact that the intersection pairing is negative definite on \(H^{\perp}\) in \(\operatorname{Num} X\).
Links to this page
#to_work