Let \(k\) be an algebraically closed field, and let \(X\subseteq\PP_k^n\) be a closed subvariety of positive dimension which is nonsingular in codimension one. For a divisor \(D=\sum n_iY_i\) on \(X\), define its degree to be \(\sum n_i\deg Y_i\), using the projective degree of each \(Y_i\) from (I, §7).
- Let \(V\) be an irreducible hypersurface in \(\PP^n\) 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 \(\PP^n\) with \(Y_i \intersect U_i \neq \emptyset\), and let \(n_i = v_{Y_i}(\bar f_i)\), where \(\bar f_i\) is the restriction of \(f_i\) to \(U_i \intersect X\).
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 \(\Div \PP^n\) consisting of divisors none of whose components contain \(X\), to \(\Div X\).
If \(D\) is a principal divisor on \(\PP^n\) for which \(D . X\) is defined as in (a), show that \(D . X\) is principal on \(X\). Thus we get a homomorphism \(\Cl \PP^n \to \Cl X\).
Show that the integer \(n_i\) defined in (a) is the same as the intersection multiplicity \(i(X,V;Y_i)\) defined in (I, §7). Then use the generalized Bezout theorem (I, 7.7) to show that for any divisor \(D\) on \(\PP^n\) none of whose components contain \(X\), \[ \deg(D . X)=(\deg D)(\deg X). \]
If \(D\) is a principal divisor on \(X\), show that there is a rational function \(f\) on \(\PP^n\) such that \(D=(f).X\). Conclude that \(\deg D = 0\), so the degree function defines a homomorphism \(\deg: \Cl X \to \ZZ\). This gives another proof of (6.10), since any complete nonsingular curve is projective. Finally, the square formed by \(\Cl \PP^n \to \Cl X\), by the degree isomorphism \(\Cl \PP^n \to \ZZ\), by \(\deg: \Cl X \to \ZZ\), and by multiplication by \(\deg X\) on \(\ZZ\) commutes. In particular, the map \(\Cl \PP^n \to \Cl X\) is injective.
(Positive dimension)
The degree formula and injectivity assertion require \(\dim X\ge1\), as in the hypothesis of Theorem I.7.7. For \(X\) a closed point of \(\PP_k^n\) with \(n\ge1\) and a hyperplane \(H\) avoiding that point, \(H.X=0\) but \((\deg H)(\deg X)=1\). The point has zero class group, so the homomorphism from \(\Cl(\PP_k^n)\cong\ZZ\) cannot be injective. The source exercise does not explicitly exclude this case [@Har10a, Exercise II.6.2].
Solution
Let \(\operatorname{Div}_X(\PP_k^n)\) be the free subgroup of \(\operatorname{Div}(\PP_k^n)\) generated by the prime hypersurfaces not containing \(X\). Divisors and class groups use the Weil-divisor convention.
1. Intersection with \(X\) defines the homomorphism in part (a).
Let \(V\) be an allowed prime hypersurface and let \(Y_i\) be a component of \(V\cap X\), with generic point \(\eta_i\). Its local ring \(R_i=\OO_{X,\eta_i}\) is a DVR because \(Y_i\) has codimension one and \(X\) is regular in codimension one. Two local equations for \(V\) differ by a unit near \(\eta_i\) in the ambient projective space. Their restrictions differ by a unit of \(R_i\), so their valuations agree. The restricted equation is nonzero because \(V\) does not contain \(X\). Thus \(n_i=v_{Y_i}(\bar f_i)\) is well-defined and finite.
There are finitely many components, since \(V\cap X\) is noetherian. The sum \(V.X=\sum_i n_iY_i\) is therefore a divisor independent of the local equations. Each divisor in \(\operatorname{Div}_X(\PP_k^n)\) has a unique finite expression in its prime hypersurfaces. Extending this assignment by \(\ZZ\)-linearity gives the asserted homomorphism.
2. Principal divisors are carried to principal divisors, and the assignment descends to \(\rho:\Cl(\PP_k^n)\to\Cl(X)\), as in part (b).
Write a nonzero rational function on \(\PP_k^n\) as \(f=F/G\), where \(F,G\) are relatively prime homogeneous polynomials of the same degree. If \(\operatorname{div}(f)\) belongs to \(\operatorname{Div}_X(\PP_k^n)\), none of the irreducible factors of \(F\) or \(G\) contains \(X\). Thus both restrictions are nonzero at the generic point of \(X\) and their quotient defines \(f_X\in K(X)^\times\). Computing on an affine chart through each prime-divisor generic point, the valuation of \(f_X\) is the difference of the valuations of the restricted numerator and denominator. By step 1, this is precisely the coefficient in \((\operatorname{div}f).X\). Hence \[ (\operatorname{div}_{\PP_k^n}f).X=\operatorname{div}_X(f_X). \]
There is a coordinate hyperplane \(H\) not containing \(X\), since some homogeneous coordinate is nonzero at its generic point. The class \([H]\) generates \(\Cl(\PP_k^n)\cong\ZZ\) [@Har10a, Proposition II.6.4]. Consequently the allowed divisor subgroup maps onto the ambient class group. Two allowed divisors representing the same class differ by an allowed principal divisor, whose intersection is principal by the preceding argument. Their intersections therefore give the same class on \(X\), defining \(\rho\).
3. The valuation coefficient equals the source intersection multiplicity, and the degree identity in part (c) follows.
Let \(V\) have homogeneous equation \(F\) of degree \(d\), and put \(C=k[x_0,\ldots,x_n]/I_X\). For a component \(Y_i\) of \(V\cap X\), let \(\eta_i\) be its generic point, put \(R_i=\OO_{X,\eta_i}\), and let \(\mathfrak m_i\) be the maximal ideal of this DVR. Let \(\mathfrak p_i\) be the homogeneous prime of \(C\) corresponding to \(Y_i\). The multiplicity \(i(X,V;Y_i)\) is the length of \((C/(F))_{\mathfrak p_i}\) over \(C_{\mathfrak p_i}\), by the definition following [@Har10a, Proposition I.7.4].
Choose a coordinate \(x_j\notin\mathfrak p_i\) and set \(B=(C_{x_j})_0\). The graded localization is \(C_{x_j}\cong B[x_j,x_j^{-1}]\). If \(\mathfrak q_i\) is the prime of \(B\) corresponding to \(\eta_i\), then \(B_{\mathfrak q_i}=R_i\) and \[ C_{\mathfrak p_i}\cong R_i[x_j,x_j^{-1}]_{\mathfrak m_iR_i[x_j,x_j^{-1}]}. \] This is a DVR with the same uniformizer \(\pi_i\) as \(R_i\). The local equation \(F/x_j^d\) equals a unit times \(\pi_i^{n_i}\) in \(R_i\), hence also in this extension DVR. Since \(x_j\) is a unit, the quotient by \(F\) has length \(n_i\) there: its successive quotients by powers of \(\pi_i\) are one-dimensional over its residue field. Therefore \(i(X,V;Y_i)=n_i\).
The generalized Bezout theorem applies because \(\dim X\ge1\) and \(V\) does not contain \(X\) [@Har10a, Theorem I.7.7]. It gives \[ \deg(V.X)=\sum_i n_i\deg Y_i=(\deg V)(\deg X). \] Extending linearly over allowed prime hypersurfaces proves the displayed identity for every allowed divisor \(D\).
4. Every principal divisor on \(X\) is the intersection with an allowed principal divisor on \(\PP_k^n\).
Let \(g\in K(X)^\times\). Choose a coordinate chart \(D_+(x_j)\) meeting \(X\). Its intersection with \(X\) is affine, with coordinate ring a quotient of \(k[x_0/x_j,\ldots,x_n/x_j]\). Write \(g=a/b\) in its fraction field, with \(a,b\) nonzero in that coordinate ring, and lift them to affine polynomials \(A_0,B_0\) in the chart coordinates. Choose an integer \(d\) at least their total degrees, and homogenize both to degree \(d\) using \(x_j\), obtaining homogeneous polynomials \(F,G\). Their restrictions are nonzero on \(X\) because their dehomogenizations are \(a,b\).
The rational function \(f=F/G\) on \(\PP_k^n\) restricts to \(g\). No irreducible factor of \(F\) or \(G\) contains \(X\): otherwise its product would vanish identically on the integral variety \(X\). Hence \(\operatorname{div}f\) is allowed, even after cancellation of common factors. Step 2 gives \[ \operatorname{div}_X(g)=(\operatorname{div}_{\PP_k^n}f).X. \]
5. The degree map descends to \(\deg_X:\Cl(X)\to\ZZ\) and satisfies \(\deg_X\circ\rho=(\deg X)\deg_{\PP_k^n}\); consequently \(\rho\) is injective.
4.Step 3 gives\[ \deg_X(\operatorname{div}_Xg) =(\deg\operatorname{div}_{\PP_k^n}f)(\deg X)=0, \]since a principal divisor on projective space has degree zero.Thus the additive degree function on divisors descends to the class group.For every ambient divisor class, choose an allowed representative as in step
2.The identity in step 3 gives the asserted equality of group homomorphisms, which is the commutativity of the required square.The degree of a nonempty projective variety is a positive integer [@Har10a, Proposition I.7.6(a)], so multiplication by \(\deg X\) on \(\ZZ\) is injective.Since \(\deg_{\PP_k^n}\) is an isomorphism, \(\rho\) must be injective.Every complete nonsingular curve is projective [@Har10a, Chapter I, §6], so the vanishing just proved also gives the principal-divisor degree assertion of II.6.10.
6. Q.E.D.
Step 1 proves part (a), step 2 proves part (b), step 3 proves part (c), and steps <1>4–<1>5 prove all assertions in part (d).