The Brauer group of an affine cone
Let \(k\) be an algebraically closed field of characteristic zero. Fixing integers \(d\) and \(\nu\) both \(\geq 1\), let \(f\) be a square-free homogeneous polynomial of degree \(d\) in \(k[x_0,\cdots, x_{\nu}]\). Let \(V=V(f)\) be the variety in \({\mathbb{P}}^{\nu}=\text{Proj} k[x_0,\cdots,x_{\nu}]\) defined by \(f\). Put \(X={\mathbb{A}}^{\nu+1}=\text{Spec} k[x_0,\cdots, x_{\nu}]\) and let \(C=Z(f)\) in \({\mathbb{A}}^{\nu+1}\) denote the affine cone over \(V\) in \(X\). Further let \(P=(0,\cdots, 0)\) denote the vertex of the cone \(C\) in \({\mathbb{A}}^{\nu+1}\) and \(\overline{C}\) the projective closure of \(C\) in \(\overline X={\mathbb{P}}^{\nu+1}\). View \(V\) as the hypersurface section of \(\overline C\) at infinity. Then \(\overline C=C\cup V\). In the paper under review, the Brauer groups of the varieties \(C\), \(C-P\), \(X-C\) and \(\overline X-\overline C\) are computed using cohomological methods (where all cohomology and sheaves are for the étale topology). First fix the necessary notations: \(B(Y)\) denotes the Brauer group of a variety \(Y\), and \(B'(Y)\) the cohomological Brauer group of \(Y\), which is \(H^2(Y,{\mathbb{G}}_m)\). For an integer \(n\geq 1\), \(\mu_n\) denotes the group of \(n\)-th roots of unity (which is non-canonically isomorphic to the group \({\mathbb{Z}}/n\)), and \(\mu_n(-1)\) stands for \(\text{Hom}_{{\mathbb{Z}}}(\mu_n,{\mathbb{Z}}/n)\). The main results are summarized as follows. Proposition 1: Let \(V\) be a reduced variety in \({\mathbb{P}}^{\nu}\) and \(C\) the cone over \(V\) in \(X={\mathbb{A}}^{\nu+1}\). Then \(B(C)=0\). Theorem 2: Let \(V\) be a nonsingular hypersurface in \({\mathbb{P}}^{\nu}\) of degree \(d\) and \(C\) the affine cone over \(V\) in \(X={\mathbb{A}}^{\nu+1}\). Let \(\overline C\) denote the projective closure of \(C\) in \({\mathbb{P}}^{\nu+1}\), and let \(P\) be the vertex of \(C\). Then the following assertions hold: (a) \(B(X-C)\simeq H^1(C-P, {\mathbb{Q}}/{\mathbb{Z}})\). In particular, if \(\nu=2\), there is a split-exact sequence: \(0\to H^1(V,{\mathbb{Q}}/{\mathbb{Z}})\to H^1(C-P,{\mathbb{Q}}/{\mathbb{Z}})\to\mu_d(-1)\to 0,\) and if \(\nu\geq 3\), \(B(X-C)=0.\) (b) \(B'(\overline C-P)\simeq B'(V)\). In particular, if \(\nu=2\), \(B(C-P)\simeq H^1(V,{\mathbb{Q}}/{\mathbb{Z}})\), and if \(\nu\geq 3\), \(B'(C-P)\simeq B'(V)\). When \(V\) is singular, the corresponding result to theorem 2 is formulated as follows. Theorem 3: Let \(V\) be a reduced effective divisor in \({\mathbb{P}}^{\nu}\) with affine cone \(C\) in \(X={\mathbb{A}}^{\nu+1}\). Let \(\widetilde C\) denote the projective closure of \(C\) in \(\widetilde X ={\mathbb{P}}^{\nu+1}\). Then the following assertions hold: (a) \(B(\widetilde X-\widetilde C)\simeq B({\mathbb{P}}^{\nu}-V).\) In particular, if \(\nu=2\), \(B(X-C)\simeq H^1({\mathbb{P}}^2-V,{\mathbb{Q}}/{\mathbb{Z}}) \oplus B({\mathbb{P}}^2-V)\), and if \(C\) is normal, \(B(X-C)\) is isomorphic to the subgroup of torsion elements in the divisor class group \(\text{Cl}(C)\) of \(C\). (b) Let \(\nu=2\). If \(V\) is an integral projective plane curve of degree \(d\) with equation \(f(x,y,z)=0\), then \(B(C-P)\simeq H^1(\widetilde V, {\mathbb{Q}}/{\mathbb{Z}})\) where \(\widetilde V\) is the normalization of \(V\).
scientific article; zbMATH DE number 1563000
scientific article