Boundedness of canonical \(\mathbb{Q}\)-Fano 3-folds. (English) Zbl 0981.14016

Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi; Takagi, Hiromichi

Proc. Japan Acad., Ser. A 76, No. 5, 73-77 (2000).

Let \(X\) be a terminal weak \(\mathbb{Q}\)-Fano 3-fold over an algebraically closed field of characteristic zero. The Gorenstein index \(I(X)\) of \(X\) is the smallest positive integer \(I\) such that \(IK_X\) is a Cartier divisor.
In this paper the authors give an effective bound for \(I(X)\) and for \((-K_X)^3\). Moreover they prove that the terminal \(\mathbb{Q}\)-Fano 3-folds are bounded (i.e. there is a morphism of scheme of finite type whose geometric fibers include all terminal \(\mathbb{Q}\)-Fano 3-folds). The main tools used in the proofs are a “gluing technique” for rational curves and a structure theorem of the cone of nef curves.

Reviewer: L.Picco Botta (Torino)

MSC:

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

Keywords:

Fano 3-fold; Gorenstein index; cone of nef curves

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