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Birational unboundedness of \({\mathbb Q}\)-Fano threefolds. (English) Zbl 1019.14005

From the introduction: We work over an algebraically closed field of characteristic zero.
Definition 1.1. Let \(X\) be a projective variety, \(X\) is said to be a \(\mathbb{Q}\)-Fano variety, if
(1) \(X\) has \(\mathbb{Q}\)-factorial log terminal singularities, and
(2) \(-K_X\) is ample.
The reader is referred to [a paper by J. McKernan [“Boundedness of log terminal Fano pairs of bounded index” (preprint: http://front.math.ncdavis.edu/math.AG/0205214)] for the standard definitions of higher-dimension geometry, such as log terminal. The main goal of this paper is to prove the following main theorem.
Theorem 1.2. The family of \(\mathbb{Q}\)-Fano threefolds with Picard number one is birationally unbounded.

MSC:

14E05 Rational and birational maps
14J30 \(3\)-folds
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
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