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Boundedness of \(\mathbb{Q}\)-Fano threefolds. (English) Zbl 0785.14024

Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 3, 439-445 (1992).
[For the entire collection see Zbl 0745.00034.]
Let \(X\) be a \(\mathbb{Q}\)-Fano variety, i.e. \(X\) is a normal projective variety with only terminal singularities and with anti-canonical Weil divisor \(-K_ X\) ample. \(X\) is said to be \(\mathbb{Q}\)-factorial if for any Weil divisor \(D\) on \(X\) there exists a positive integer \(m\) such that \(mD\) is a Cartier divisor. The smallest positive integer \(r=r(X)\) such that \(rK_ X\) is a Cartier divisor is called the singularity index of \(X\). In the paper under review the author proves that for an arbitrary 3- dimensional \(\mathbb{Q}\)-factorial \(\mathbb{Q}\)-Fano variety \(X\) with Picard number \(\rho(X)=1\) defined over a field of characteristic 0 the singularity index \(r(X)\) and the degree \((-K_ X)^ 3\) are bounded.
If the \(\mathbb{Q}\)-Fano variety is smooth then, as the author pointed out, H. Tsuji has proved their boundedness using differential geometric methods [H. Tsuji, “Boundedness of the degree of Fano manifolds with \(b_ 2=1\)” (preprint)].

MSC:

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14J30 \(3\)-folds

Citations:

Zbl 0745.00034
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