Centrally symmetric generators in toric Fano varieties
This paper tries to reduce the classification of toric Fano varieties to that of lower dimensional ones (as base varieties) by describing clearly the base and the fibre. To be precise, let \(X\) be a smooth, complete toric variety of dimension \(n\), \(N = \Hom_{\mathbb{Z}} (\mathbb{C}^*, T)\) (with \(T\) the maximum torus in \(X\)), \(N_{\mathbb{Q}} = N \otimes_{\mathbb{Z}} \mathbb{Q}\) and \(\Sigma_X\) the fan of \(X\) in \(N_{\mathbb{Q}}\). One defines the set \(G(\Sigma_X)\) of generators of \(\Sigma_X\) as \(\{x \in N \mid x \neq 0, \langle x \rangle \in \Sigma_X\) and \(x\) is primitive in \(\langle x \rangle \cap N\}\). The author proves: Let \(X\) be a smooth toric Fano variety. Suppose that \(G(\Sigma_X)\) contains \(r > 0\) pairs of centrally symmetric generators \(\{x, -x\}\), and let \(H\) be the linear subspace of \(N_{\mathbb{Q}}\) spanned by all such pairs. (1) If \(\dim H < r\), then \(X\) is a toric bundle over a lower dimensional toric Fano variety with fibre a product of del Pezzo varieties. (2) If \(\dim H = n\), then \(X\) is a product of del Pezzo varieties, pseudo del Pezzo varieties and a power of the projective line [the last result extends \textit{G. Ewald}'s earlier result: Discrete Comput. Geom. 3, 49--54 (1988; Zbl 0686.14042)]. The author has other similar results for \(\dim H = n-1\), etc. Readers are advised to consult the paper.
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