Extremal rays on higher dimensional projective varieties
The paper under review investigates small extremal contractions \(f\) of a smooth projective variety \(X\) of dimension \(n=2k\) under the following assumption. Suppose that each irreducible component \(E_ i\) of the exceptional locus \(E\) of \(f\) is a smooth subvariety of dimension \(k\). Theorem 1 states that then \(E_ i\cong\mathbb{P}^ k\). Note that always, by \textit{J. A. Wiśniewski} [J. Reine Angew. Math. 417, 141-157 (1991; Zbl 0721.14023)], \(\dim E\geq k\). So, the considered case is extreme. But the assumption that \(E_ i\)'s are smooth is very strong. Under the last assumption and for \(X\) of dimension 5, the author proves that \(E_ i\cong\mathbb{P}^ 3\), or \(\mathbb{Q}^ 3\) (a 3-dimensional hyperquadric), or a linear \(\mathbb{P}^ 2\)-bundle over a smooth curve.
scientific article; zbMATH DE number 59967
scientific article