2_6x Divisors

II.6: Divisors

In this section we will consider schemes satisfying the following condition:

\((*) X\) is a noetherian integral separated scheme which is regular in codimension one.

II.6.1. #to_work

Let \(X\) be a scheme satisfying \((*)\). Then \(X \times \mathbf{P}^n\) also satisfies \((*)\), and \(\operatorname{Cl} \left(X \times \mathbf{P}^n\right) \cong( \operatorname{Cl} X) \times \mathbf{Z}\)

II.6.2. * Varieties in Projective Space. #to_work

Let \(k\) be an algebraically closed field, and let \(X\) be a closed subvariety of \(\mathbf{P}_k^n\) which is nonsingular in codimension one (hence satisfies \((*)\) ). For any divisor \(D=\sum n_i Y_i\) on \(X\), we define the degree of \(D\) to be \(\sum n_i \operatorname{deg} Y_i\), where \(\operatorname{deg} Y_i\) is the degree of \(Y_i\), considered as a projective variety itself (I, §7).

• Let \(V\) be an irreducible hypersurface in \(\mathbf{P}^n\) which does not contain \(X\), and let \(Y_i\) be the irreducible components of \(V \cap X\). They all have codimension 1 by (I, Ex. 1.8). For each \(i\), let \(f_i\) be a local equation for \(V\) on some open set \(U_i\) of \(\mathbf{P}^n\) for which \(Y_i \cap U_i \neq \varnothing\), and let \(n_i=v_{Y_i}(\mkern 1.5mu\overline{\mkern-1.5muf_i\mkern-1.5mu}\mkern 1.5mu)\), where \(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu_i\) is the restriction of \(f_i\) to \(U_i \cap X\).

  Then we define the divisor \(V . X\) to be \(\sum n_i Y_i\). Extend by linearity, and show that this gives a well-defined homomorphism from the subgroup of Div \(\mathbf{P}^n\) consisting of divisors, none of whose components contain \(X\), to \(\operatorname{Div} X\).

• If \(D\) is a principal divisor on \(\mathbf{P}^n\), for which \(D . X\) is defined as in (a), show that \(D . X\) is principal on \(X\). Thus we get a homomorphism \(\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X\).

• Show that the integer \(n_i\) defined in (a) is the same as the intersection multiplicity \(i\left(X, V ; Y_i\right)\) defined in \((\mathrm{I}, \S 7)\). Then use the generalized Bézout theorem \((\mathrm{I}, 7.7)\) to show that for any divisor \(D\) on \(\mathbf{P}^n\), none of whose components contain \(X\), \begin{align*} \operatorname{deg}(D \cdot X)=(\operatorname{deg} D) \cdot(\operatorname{deg} X) . \end{align*}

• If \(D\) is a principal divisor on \(X\), show that there is a rational function \(f\) on \(\mathbf{P}^n\) such that \(D=(f) . X\). Conclude that \(\operatorname{deg} D=0\). Thus the degree function defines a homomorphism deg: \(\operatorname{Cl} X \rightarrow \mathbf{Z}\). ¹ Finally, there is a commutative diagram

Link to Diagram

and in particular, we see that the map \(\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X\) is injective.

II.6.3. * Cones. #to_work

In this exercise we compare the class group of a projective variety \(V\) to the class group of its cone (I, Ex. 2.10). So let \(V\) be a projective variety in \(\mathbf{P}^n\), which is of dimension \(\geqslant 1\) and nonsingular in codimension 1 . Let \(X=C(V)\) be the affine cone over \(V\) in \(\mathbf{A}^{n+1}\), and let \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) be its projective closure in \(\mathbf{P}^{n+1}\). Let \(P \in X\) be the vertex of the cone.

• Let \(\pi: \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P \rightarrow V\) be the projection map. Show that \(V\) can be covered by open subsets \(U_i\) such that \(\pi^{-1}\left(U_i\right) \cong U_i \times \mathbf{A}^1\) for each \(i\), and then show as in (6.6) that \(\pi^*: \operatorname{Cl} V \rightarrow \operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)\) is an isomorphism. Since \(\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \cong\) \(\operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)\), we have also \(\operatorname{Cl} V \cong \operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\).

• We have \(V \subseteq \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) as the hyperplane section at infinity. Show that the class of the divisor \(V\) in \(\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) is equal to \(\pi^*\) (class of \(V . H\) ) where \(H\) is any hyperplane of \(\mathbf{P}^n\) not containing \(V\). Thus conclude using \((6.5)\) that there is an exact sequence \begin{align*} 0 \rightarrow \mathbf{Z} \rightarrow \operatorname{Cl} V \rightarrow \operatorname{Cl} X \rightarrow 0, \end{align*} where the first arrow sends \(1 \mapsto V . H\), and the second is \(\pi^*\) followed by the restriction to \(X-P\) and inclusion in \(X\). (The injectivity of the first arrow follows from the previous exercise.)

• Let \(S(V)\) be the homogeneous coordinate ring of \(V\) (which is also the affine coordinate ring of \(X\) ). Show that \(S(V)\) is a unique factorization domain if and only if

  • \(V\) is projectively normal (Ex. 5.14), and
  • \(\operatorname{Cl} V \cong \mathrm{Z}\) and is generated by the class of \(V . H\).

• Let \({\mathcal{O}}_P\) be the local ring of \(P\) on \(X\). Show that the natural restriction map induces an isomorphism \(\operatorname{Cl} X \rightarrow \operatorname{Cl} \left(\operatorname{Spec}{\mathcal{O}}_P\right)\).

II.6.4. #to_work

Let \(k\) be a field of characteristic \(\neq 2\). Let \(f \in k\left[x_1, \ldots, x_n\right]\) be a square-free nonconstant polynomial, i.e., in the unique factorization of \(f\) into irreducible polynomials, there are no repeated factors. Let \(A=k\left[x_1, \ldots, x_n, z\right] /\left(z^2-f\right)\). Show that \(A\) is an integrally closed ring. ²

Conclude that \(A\) is the integral closure of \(k\left[x_1, \ldots, x_n\right]\) in \(K\).

II.6.5. * Quadric Hypersurfaces. #to_work

Let char \(k \neq 2\), and let \(X\) be the affine quadric hypersurface ³ \begin{align*} \operatorname{Spec} k\left[x_0, \ldots, x_n\right] /\left(x_0^2+x_1^2+\ldots+x_r^2\right) .\end{align*}

• Show that \(X\) is normal if \(r \geqslant 2\) (use (Ex. 6.4)).

• Show by a suitable linear change of coordinates that the equation of \(X\) could be written as \(x_0 x_1=x_2^2+\ldots+x_r^2\). Now imitate the method of \((6.5 .2)\) to show that:

  • If \(r=2\), then \(\operatorname{Cl} X \cong \mathbf{Z} / 2 \mathbf{Z}\);
  • If \(r=3\), then \(\operatorname{Cl} X \cong \mathbf{Z}\) (use (6.6.1) and (Ex. 6.3) above);
  • If \(r \geqslant 4\) then \(\operatorname{Cl} X=0\).

• Now let \(Q\) be the projective quadric hypersurface in \(\mathbf{P}^n\) defined by the same equation. Show that:

  • If \(r=2, \operatorname{Cl} Q \cong \mathbf{Z}\), and the class of a hyperplane section \(Q . H\) is twice the generator;
  • If \(r=3, \operatorname{Cl} Q \cong \mathbf{Z} \oplus \mathbf{Z}\);
  • If \(r \geqslant 4, \operatorname{Cl} Q \cong \mathbf{Z}\), generated by \(Q . H\).

• Prove Klein’s theorem, which says that if \(r \geqslant 4\), and if \(Y\) is an irreducible subvariety of codimension 1 on \(Q\), then there is an irreducible hypersurface \(V \subseteq \mathbf{P}^n\) such that \(V \cap Q=Y\), with multiplicity one. In other words, \(Y\) is a complete intersection.

  (First show that for \(r \geqslant 4\), the homogeneous coordinate ring \(S(Q)=k\left[x_0, \ldots, x_n\right] /\left(x_0^2+\ldots+x_r^2\right)\) is a UFD.)

II.6.6. #to_work

Let \(X\) be the nonsingular plane cubic curve \(y^2 z=x^3-x z^2\) of \((6.10 .2)\).

• Show that three points \(P, Q, R\) of \(X\) are collinear if and only if \(P+Q+R=0\) in the group law on \(X\).

  Note that the point \(P_0=(0,1,0)\) is the zero element in the group structure on \(X\).

• A point \(P \in X\) has order 2 in the group law on \(X\) if and only if the tangent line at \(P\) passes through \(P_0\).

• A point \(P \in X\) has order 3 in the group law on \(X\) if and only if \(P\) is an inflection point. ⁴

• Let \(k=\mathbf{C}\). Show that the points of \(X\) with coordinates in \(\mathbf{Q}\) form a subgroup of the group \(X\). Can you determine the structure of this subgroup explicitly?

II.6.7. * #to_work

Let \(X\) be the nodal cubic curve \(y^2 z=x^3+x^2 z\) in \(\mathbf{P}^2\). Imitate (6.11.4) and show that the group of Cartier divisors of degree \(0, \mathrm{CaCl}^{\circ} X\), is naturally isomorphic to the multiplicative group \(\mathbf{G}_m\).

II.6.8. #to_work

• Let \(f: X \rightarrow Y\) be a morphism of schemes. Show that \(\mathcal{L} \mapsto f^* \mathcal{L}\) induces a homomorphism of Picard groups, \(f^*:\operatorname{Pic}Y \rightarrow \operatorname{Pic}X\).

• If \(f\) is a finite morphism of nonsingular curves, show that this homomorphism corresponds to the homomorphism \(f^*: \operatorname{Cl} Y \rightarrow \operatorname{Cl} X\) defined in the text, via the isomorphisms of (6.16).

• If \(X\) is a locally factorial integral closed subscheme of \(\mathbf{P}_k^n\), and if \(f: X \rightarrow \mathbf{P}^n\) is the inclusion map, then \(f^*\) on Pic agrees with the homomorphism on divisor class groups defined in (Ex. 6.2) via the isomorphisms of (6.16).

II.6.9. * Singular Curves. #to_work

Here we give another method of calculating the Picard group of a singular curve. Let \(X\) be a projective curve over \(k\), let \(\tilde{X}\) be its normalization, and let \(\pi: \tilde{X} \rightarrow X\) be the projection map (Ex. 3.8). For each point \(P \in X\), let \({\mathcal{O}}_P\) be its local ring, and let \(\tilde{{\mathcal{O}}}_P\) be the integral closure of \({\mathcal{O}}_P\). We use a \(*\) to denote the group of units in a ring.

• Show there is an exact sequence ⁵ \begin{align*} 0 \rightarrow \bigoplus_{P \in X} \tilde{{\mathcal{O}}}_P^* / {\mathcal{O}}_P^* \rightarrow \operatorname{Pic}X \stackrel{\pi^*}{\rightarrow} \operatorname{Pic}\tilde{X} \rightarrow 0 . \end{align*}
• Use (a) to give another proof of the fact that if \(X\) is a plane cuspidal cubic curve, then there is an exact sequence \begin{align*} 0 \rightarrow \mathbf{G}_a \rightarrow \operatorname{Pic}X \rightarrow \mathbf{Z} \rightarrow 0, \end{align*} and if \(X\) is a plane nodal cubic curve, there is an exact sequence \begin{align*} 0 \rightarrow \mathbf{G}_m \rightarrow \operatorname{Pic}X \rightarrow \mathbf{Z} \rightarrow 0 \end{align*}

II.6.10. The Grothendieck Group \(K(X)\). #to_work

Let \(X\) be a noetherian scheme. We define \(K(X)\) to be the quotient of the free abelian group generated by all the coherent sheaves on \(X\), by the subgroup generated by all expressions \({\mathcal{F}}-{\mathcal{F}}^{\prime}-{\mathcal{F}}^{\prime \prime}\), whenever there is an exact sequence \(0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0\) of coherent sheaves on \(X\). If \({\mathcal{F}}\) is a coherent sheaf, we denote by \(\gamma({\mathcal{F}})\) its image in \(K(X)\).

• If \(X=\mathbf{A}_k^1\), then \(K(X) \cong \mathbf{Z}\).

• If \(X\) is any integral scheme, and \({\mathcal{F}}\) a coherent sheaf, we define the rank of \({\mathcal{F}}\) to be \(\operatorname{dim}_K {\mathcal{F}}_{\xi}\), where \(\xi\) is the generic point of \(X\), and \(K={\mathcal{O}}_{\xi}\) is the function field of \(X\).

  Show that the rank function defines a surjective homomorphism rank: \(K(X) \rightarrow \mathbf{Z}\).

• If \(Y\) is a closed subscheme of \(X\), there is an exact sequence \begin{align*} K(Y) \rightarrow K(X) \rightarrow K(X-Y) \rightarrow 0, \end{align*} where the first map is extension by zero, and the second map is restriction. ⁶

II.6.11. *The Grothendieck Group of a Nonsingular Curve. #to_work

Let \(X\) be a nonsingular curve over an algebraically closed field \(k\). We will show that \(K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}\), in several steps.

• For any divisor \(D=\sum n_i P_i\) on \(X\), let \(\psi(D)=\sum n_i \gamma\left(k\left(P_i\right)\right) \in K(X)\), where \(k\left(P_i\right)\) is the skyscraper sheaf \(k\) at \(P_i\) and 0 elsewhere. If \(D\) is an effective divisor, let \({\mathcal{O}}_D\) be the structure sheaf of the associated subscheme of codimension 1, and show that \(\psi(D)=\gamma\left({\mathcal{O}}_D\right)\). Then use (6.18) to show that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D\), so \(\psi\) defines a homomorphism \(\psi: \operatorname{Cl} X \rightarrow K(X)\)

• For any coherent sheaf \({\mathcal{F}}\) on \(X\), show that there exist locally free sheaves \(\mathcal{E}_0\) and \(\mathcal{E}_1\) and an exact sequence \(0 \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow {\mathcal{F}}\rightarrow 0\). Let \(r_0=\) rank \(\mathcal{E}_0\), \(r_1=\operatorname{rank} \mathcal{E}_1\), and define \begin{align*} \operatorname{det}{\mathcal{F}}= \left(\bigwedge^{r_0} \mathcal{E}_0\right) \otimes\left(\bigwedge^{r_1} \mathcal{E}_1\right)^{-1} \in \operatorname{Pic}X .\end{align*} Here \(\bigwedge\) denotes the exterior power (Ex. 5.16). Show that \(\operatorname{det}{\mathcal{F}}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow \operatorname{Pic}X\). Finally show that if \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathcal{L}(D)\).

• If \({\mathcal{F}}\) is any coherent sheaf of rank \(r\), show that there is a divisor \(D\) on \(X\) and an exact sequence \begin{align*} 0 \rightarrow \mathcal{L}(D)^{\oplus r} \rightarrow {\mathcal{F}}\rightarrow \mathcal{T} \rightarrow 0 ,\end{align*} where \(\mathcal{T}\) is a torsion sheaf. Conclude that if \({\mathcal{F}}\) is a sheaf of rank \(r\), then \begin{align*} \gamma({\mathcal{F}})-r \gamma\left({\mathcal{O}}_X\right) \in \operatorname{Im} \psi .\end{align*}

• Using the maps \(\psi, \operatorname{det}, \mathrm{rank}\), and \(1 \mapsto \gamma\left({\mathcal{O}}_X\right)\) from \(\mathbf{Z} \rightarrow K(X)\), show that \(K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}\).

II.6.12.

Let \(X\) be a complete nonsingular curve. Show that there is a unique way to define the degree of any coherent sheaf on \(X\), deg \({\mathcal{F}}\in \mathbf{Z}\), such that: #to_work

• If \(D\) is a divisor, \(\operatorname{deg} \mathcal{L}(D)=\operatorname{deg} D\);
• If \({\mathcal{F}}\) is a torsion sheaf(meaning a sheaf whose stalk at the generic point is zero), then \(\operatorname{deg} {\mathcal{F}}=\sum_{P \in X}\) length \(\left({\mathcal{F}}_P\right)\); and

• If \(0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow \mathcal{\mathcal { F }} \rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0\) is an exact sequence, then \(\operatorname{deg} {\mathcal{F}}=\operatorname{deg} {\mathcal{F}}^{\prime}+\) \(\operatorname{deg} {\mathcal{F}}^{\prime \prime}\).