On the birational unboundedness of higher dimensional \({\mathbb{Q}}\)-Fano varieties
type: Item
instance of: scholarly article
title: On the birational unboundedness of higher dimensional \({\mathbb{Q}}\)-Fano varieties
zbMATH Open document ID: 1174.14033
author: Takuzo Okada
published in: Mathematische Annalen
publication date: 2009-08-27T00:00:00Z
review text: 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.
reviewed by: Jing Zhang
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J40
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 5598901
MaRDI profile type: Publication
full work available at URL: s00208-009-0351-7
OpenAlex ID: W1590204741
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DOI: 10.1007/S00208-009-0351-7
description: scientific article; zbMATH DE number 5598901