On the birational unboundedness of higher dimensional \({\mathbb{Q}}\)-Fano varieties
A normal projective variety defined over the field \(\mathbb{C}\) of complex numbers is a \({\mathbb{Q}}\)-Fano variety if it is \(\mathbb{Q}\)-factorial, has only log terminal singularities and its anticanonical divisor is ample. A class of varieties \(\mathcal{B}\) is birationally bounded if there exists a morphism \(f: {\mathcal{X}}\rightarrow \mathcal{S}\) between algebraic schemes such that every variety in \(\mathcal{B}\) is birational to one of the geometric fibers of \(f\). \(\mathcal{B}\) is birationally unbounded if it is not birationally bounded. In this paper, the author proves the following main theorem: If \(n\geq 6\), then the family of \(\mathbb{Q}\)-Fano \(n\)-folds defined over \(\mathbb{C}\) with Picard number one is birationally unbounded. The author constructed non-ruled \(\mathbb{Q}\)-Fano weighted hypersurfaces with Picard number one in a different paper [Nagoya Math. J. 194, 1--32 (2009; Zbl 1193.14018)]. The idea of the proof is to show that these examples are birationally unbounded if the dimension is greater than or equal to 6.
scientific article; zbMATH DE number 5598901