Completion of real fans and Zariski-Riemann spaces
A well-known theorem due to \textit{M. Nagata} [J. Math. Kyoto Univ. 2, 1--10 (1962; Zbl 0109.39503); ibid. 3, 89--102 (1963; Zbl 0223.14011)] asserts that if \(X\) is any (noetherian) scheme over an algebraically closed field \(k\), then \(X\) can be imbedded as an open subscheme of a complete variety \(\bar{X}\). By a general result of \textit{H. Sumihiro} [J. Math. Kyoto Univ. 14, 1--28 (1974; Zbl 0277.14008)] any toric variety can be equivariantly imbedded as an open subvariety of a complete toric variety. The authors give two different proofs of this result. Let \(N\cong \mathbb Z^r\) be a lattice in \(N_\mathbb R:=N\otimes\mathbb R\). Corresponding to any toric variety \(X\) of dimension \(r\), one has a fan \(\Sigma\) consisting of certain strongly convex rational polyhedral cones in \(N_\mathbb R\). The variety \(X\) is complete if and only if \(\Sigma\) is complete, that is, the support of \(\Sigma \) equals \(N_\mathbb R\). The authors give a purely combinatorial proof that any (finite) real fan, is a subfan of a complete real fan. By a real fan is meant a fan whose elements need not be rational polyhedral cones. The second proof uses the notion of Zariski-Riemann space \(ZR(\Sigma)\) (resp. \(ZR(M)\)) of a fan \(\Sigma\) (resp. the dual lattice \(M:=N^\vee\)). It is shown that the spaces \(ZR(\Sigma)\) and \(ZR(\Sigma)\) are quasi-compact (Theorem 2.5) and that the fan \(\Sigma\) is complete if and only if \(ZR(\Sigma)=ZR(M)\). Starting with a fan \(\Sigma\) which is not complete, and a \(v\in ZR(M)\) which is not in \(ZR(\Sigma)\), it is shown that there exists a fan \(\Sigma_v\) which contains \(\Sigma\) and \(v\in ZR(\Sigma_v)\). Using these results it is shown that \(\Sigma\) can be imbedded in a complete fan \(\bar{\Sigma}\).
scientific article; zbMATH DE number 5076981
scientific article