A Fano 3-fold with nonrational singularities and a two-dimensional basis
Let X be a normal complex Fano 3-fold, and let \(\Sigma\) be the locus of non-rational singular points of \(X\); in particular \(\dim \Sigma \leq 1\). In a previous paper the author shows that if \(\dim \Sigma = 0\) (or, more general, if \(X\) has isolated non-rational singular points) then \(X\) is isomorphic to a projective cone over a possibly singular K3 or abelian surface. Now the author studies the remaining case when \(\Sigma\) is of pure dimension 1. Then \(X\) has always \({\mathbb{Q}}\)-factorial terminal modifications \(Y\); and, after the choice of such particular \(Y_o\), the structure of \(X\) should depend on the dimension (= 0,1,2) of the base \(Z_o\) of this particular modification (see pp. 3785-3786). The main result of this paper, theorem 3, treats the case \(\dim Z_o = 2\): If the non-rational singular locus \(\Sigma\) of the normal complex Fano 3-fold \(X\) is of pure dimension 1, and if \(X\) admits a \({\mathbb{Q}}\)-factorial terminal modification \(Y_o\) with a \(2\)-dimensional base \(Z_o\), then \(X\) has a \({\mathbb{Q}}\)-factorial terminal modification \(Y\) with a smooth 2-dimensional base \(Z\) such that the extremal contraction \(Y \rightarrow Z\) is a \({\mathbb{P}}^1\)-bundle. In short, such \(X\) must be naturally birational to a \({\mathbb{P}}^1\)-bundle over a smooth surface. The study in sections 4 and 5 gives a satisfactory description of the possible cases of projective bundles \(Y \rightarrow Z\) which may come from a Fano 3-fold \(X\) as in the conclusion of theorem 3.
\(\mathbb{Q}\)-factorial terminal modification
scientific article; zbMATH DE number 1017603