Let \(k\) be an algebraically closed field, and let \(V\subseteq\PP_k^n\) be a projective variety of positive dimension which is nonsingular in codimension one. Let \(X=C(V)\subseteq\AA_k^{n+1}\) be its affine cone (I, Ex. 2.10), and let \(\bar X\subseteq\PP_k^{n+1}\) be its projective closure. Let \(P \in X\) be the vertex of the cone.
Let \(\pi:\bar X\setminus\{P\}\to V\) be the projection map. Show that \(V\) has an open cover \((U_i)\) for which \(\pi^{-1}(U_i)\cong U_i\times\AA_k^1\) over \(U_i\). Then show, as in (6.6), that \(\pi^*:\Cl(V)\to\Cl(\bar X\setminus\{P\})\) is an isomorphism. Since \(\Cl(\bar X)\cong\Cl(\bar X\setminus\{P\})\), conclude that \(\Cl(V)\cong\Cl(\bar X)\).
Regard \(V\subseteq\bar X\) as the hyperplane section at infinity. Show that \([V]=\pi^*[V.H]\) in \(\Cl(\bar X)\), where \(H\subseteq\PP_k^n\) is any hyperplane not containing \(V\) and the pullback uses the identification in (a). Deduce from (6.5) the exact sequence \[ 0\longrightarrow\ZZ\xrightarrow{1\mapsto[V.H]}\Cl(V)\longrightarrow\Cl(X)\longrightarrow0. \] The second map is \(\pi^*\) followed by restriction to \(X\setminus\{P\}\) and the inverse of \(\Cl(X)\to\Cl(X\setminus\{P\})\). Prove injectivity of the first map using the degree map of the preceding exercise.
Let \(S(V)\) be the homogeneous coordinate ring of \(V\), 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 \(\Cl(V)\cong\ZZ\) is generated by \([V.H]\).
Let \(\OO_{X,P}\) be the local ring at the vertex. Show that restriction induces an isomorphism \(\Cl(X)\to\Cl(\Spec\OO_{X,P})\).
Solution
Write \(S=S(V)=k[x_0,\ldots,x_n]/I_V\), so \(X=\Spec S\) and \(\bar X=\Proj S[z]\). Then \(P=[0:\cdots:0:1]\), the section at infinity is \(V_\infty=V(z)\cong V\), and \(X=D_+(z)\). All class groups use the Weil-divisor convention.
1. If \(W\) is an integral variety regular in codimension one and \(p:Y\to W\) is a Zariski locally trivial \(\AA^1\)-bundle, then \(Y\) is regular in codimension one and \(p^*:\Cl(W)\to\Cl(Y)\) is an isomorphism.
Each trivializing chart has the form \(U\times\AA^1\). These charts are integral and regular in codimension one by [@Har10a, Proposition II.6.6]. Any two nonempty trivializing charts meet over the generic point of \(W\), so \(Y\) is integral. The local prime-divisor classification in that proposition applies on every chart. Thus a prime divisor of \(Y\) is either vertical, of the form \(p^{-1}(D)\) for a prime divisor \(D\subseteq W\), or horizontal, dominating \(W\).
At the generic point of \(p^{-1}(D)\), a uniformizer of \(\OO_{W,\eta_D}\) remains a uniformizer. At a horizontal prime divisor, every nonzero element of \(K(W)\) is a unit. Consequently \(p^*D=p^{-1}(D)\) extends linearly to divisors and satisfies \[ \operatorname{div}_Y(p^*g)=p^*(\operatorname{div}_W g) \qquad(g\in K(W)^\times). \] It therefore induces a homomorphism on class groups.
Choose a coordinate \(t\) on the generic fiber \(Y_{K(W)}\cong\AA^1_{K(W)}\). A horizontal prime divisor \(Z\) determines a prime ideal \((f)\) of \(K(W)[t]\), with \(f\) irreducible. Viewed as a rational function on \(Y\), \(f\) has horizontal divisor exactly \(Z\), with coefficient one. Thus \(Z-\operatorname{div}_Y(f)\) is vertical, proving surjectivity.
If \(p^*D=\operatorname{div}_Y(g)\), then the divisor of \(g\) on \(\AA^1_{K(W)}\) is zero. A quotient of relatively prime polynomials in \(K(W)[t]\) has zero divisor only when both polynomials are constant. Hence \(g\in K(W)^\times\), and the vertical valuations give \(D=\operatorname{div}_W(g)\). This proves injectivity.
2. The projection in part (a) is locally trivial and induces \(\Cl(V)\cong\Cl(\bar X)\).
For each nonempty \(U_i=V\cap D_+(x_i)\), put \(B_i=(S_{x_i})_0\). Projection is \([x_0:\cdots:x_n:z]\mapsto[x_0:\cdots:x_n]\), and its inverse image of \(U_i\) has coordinate ring \[ (S[z]_{x_i})_0=B_i[z/x_i]. \] Thus \(\pi^{-1}(U_i)\cong U_i\times\AA_k^1\), with fiber coordinate \(t_i=z/x_i\). The coordinates satisfy \(t_j=(x_i/x_j)t_i\) on overlaps. Step 1 now gives \(\pi^*:\Cl(V)\xrightarrow{\cong}\Cl(\bar X\setminus\{P\})\).
The charts show that \(\dim\bar X=\dim V+1\ge2\) and that \(\bar X\setminus\{P\}\) is regular in codimension one. The closed point \(P\) has codimension at least two, so \(\bar X\) and its open subset \(X\) are regular in codimension one as well. Deleting \(P\) changes neither their prime-divisor groups nor their valuations. Hence restriction identifies \(\Cl(\bar X)\) with \(\Cl(\bar X\setminus\{P\})\), and \(\Cl(X)\) with \(\Cl(X\setminus\{P\})\) [@Har10a, Proposition II.6.5(b)]. This proves part (a).
3. The hyperplane class \(h=[V.H]\) has infinite order, and part (b) gives\[ \boxed{\Cl(X)\cong\Cl(V)/\ZZ h}. \]
1.These divisors have no common component.There are no prime divisors supported at \(P\), so\[ \operatorname{div}_{\bar X}(z/\ell)=V_\infty-\pi^*(V.H). \]This proves \([V_\infty]=\pi^*h\) under the identification of step
2.
2.The result is the sequence in part (b), with the stated maps.The degree map satisfies \(\deg(h)=\deg V>0\).Thus \(mh=0\) implies \(m\deg V=0\), hence \(m=0\), proving injectivity at the left end and the displayed quotient description.
4. The unique-factorization criterion in part (c) holds.
The finitely generated domain \(S\) is noetherian. By the definition of projective normality, \(V\) is projectively normal precisely when \(S\) is integrally closed. A noetherian integrally closed domain is a unique factorization domain precisely when its divisor class group is zero [@Har10a, Proposition II.6.2]. Every unique factorization domain is integrally closed, so \[ S\text{ is a unique factorization domain} \quad\Longleftrightarrow\quad S\text{ is integrally closed and }\Cl(X)=0. \] Step 3 identifies the last vanishing with \(\Cl(V)=\ZZ h\). Since \(h\) has infinite order, this is exactly the asserted infinite cyclic group with its specified generator.
5. Restriction \(\rho:\Cl(X)\to\Cl(\Spec\OO_{X,P})\) is surjective, and its kernel is generated by the classes of prime divisors of \(X\) not containing \(P\).
Let \(\mathfrak m=(x_0,\ldots,x_n)\subseteq S\), so \(\OO_{X,P}=S_{\mathfrak m}\). The height-one primes of \(S_{\mathfrak m}\) correspond to the height-one primes of \(S\) contained in \(\mathfrak m\), and their local DVRs are unchanged. Thus restriction of divisors discards exactly the prime divisors not containing \(P\), and it carries principal divisors to principal divisors. Every local prime divisor is the restriction of its closure in \(X\), proving surjectivity.
If a divisor \(D\) becomes principal locally, choose \(g\in K(X)^\times\) whose local divisor is its restriction. Then \(D-\operatorname{div}_X(g)\) is supported on prime divisors not containing \(P\). Conversely, every such divisor restricts to zero. This proves the kernel description.
6. Every prime divisor \(Z\subseteq X\) not containing \(P\) has zero class in \(\Cl(X)\).
Let \(p:T\to V\) be the total space of the tautological line bundle \(\OO_V(-1)\). Concretely, its points are pairs \((x,[v])\in\AA_k^{n+1}\times V\) for which \(x\) lies on the line represented by \([v]\). Over \(U_i\), it has coordinate ring \(B_i[u_i]\), and its projection \(q:T\to X\) is given by \[ x_i=u_i,\qquad x_j=u_i(x_j/x_i)\quad(j\ne i), \] where \(x_j/x_i\) on the right denotes the coordinate function on \(U_i\). These formulas glue and satisfy the homogeneous equations of \(X\). The zero section \(E\cong V\) is \(q^{-1}(P)\), and \(q\) identifies \(T\setminus E\) with \(X\setminus\{P\}\); its inverse there is \(x\mapsto(x,[x])\).
The inverse image \(\widetilde Z=q^{-1}(Z)\) is therefore a prime divisor of \(T\) disjoint from \(E\). Let \(W\subseteq V\) be the regular locus and put \(T_W=p^{-1}(W)\). The regular locus is open for a variety [@Har10a, Theorem I.5.3], and \(V\setminus W\) has codimension at least two by hypothesis. Local triviality shows that \(T\setminus T_W\) also has codimension at least two. The variety \(T_W\) is regular, since locally it is the product of the regular variety \(W\) with \(\AA_k^1\).
1.For the zero section \(s:W\to T_W\), the identity \(p_Ws=\operatorname{id}_W\) shows that \(s^*\) is the inverse isomorphism.
The divisor \(\widetilde Z|_{T_W}\) is Cartier and disjoint from \(s(W)\). Its associated invertible sheaf therefore restricts to the trivial sheaf on \(s(W)\): the canonical section is nowhere zero there. Injectivity of \(s^*\) gives \[ [\OO_{T_W}(\widetilde Z|_{T_W})]=0\quad\text{in }\Pic(T_W). \] Hence \([\widetilde Z|_{T_W}]=0\) in \(\Cl(T_W)\). Step 1 ensures that \(T\) is regular in codimension one, so deleting \(T\setminus T_W\) preserves its class group by Proposition II.6.5(b). It follows that \([\widetilde Z]=0\) in \(\Cl(T)\). Restrict to \(T\setminus E\), identify it with \(X\setminus\{P\}\), and use step 2 to conclude \([Z]=0\) in \(\Cl(X)\).
7. Q.E.D.
Step 2 proves part (a), step 3 proves part (b), and step 4 proves part (c). Steps 5 and 6 make \(\rho\) surjective with zero kernel, proving the isomorphism in part (d).