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On algebraic fiber space structures on a Calabi-Yau 3-fold. Appendix by Noburu Nakayama. (English) Zbl 0793.14030
In this paper the author proves a structure theorem for minimal Calabi- Yau threefolds (i.e. $$\mathbb{Q}$$-factorial, normal, projective 3-folds with only terminal singularities, trivial canonical class, $$c_ 2 \neq 0$$, and $$\pi_ 1^{\text{alg}}=\{1\})$$. More precisely, an appropriate multiple $$m.L$$ of any nef and effective divisor $$L$$ on a minimal Calabi- Yau threefold $$X$$ gives a fibration $$\varphi:X \to W$$, and the main theorem in the paper explicitly describes the structure of the general fiber of $$\varphi$$, and the type of $$W$$, depending on the invariants $$\kappa (X,L)=\nu (X,L)=\dim W$$ and $$L.c_ 2 (X)$$. In addition, the author constructs the first known example of a Calabi-Yau threefold “of type II$$_ 0$$” (i.e. $$\nu (X,L)=2$$, $$L.c_ 2 (X)=0)$$. As an application, a simpler, and a little generalized proof of the theorem of Peternell for the existence of a rational curve on a Calabi-Yau threefold, which contains a nonzero and nonample effective divisor, is given.
The appendix, written by N. Nakayama, is devoted to the existence of families of rational curves on certain Calabi-Yau threefolds admitting a fibering structure. In connection with this, a theorem, which describes the structure of such threefolds having no families of rational curves, is proved.
Reviewer: A.Iliev (Sofia)

##### MSC:
 14J30 $$3$$-folds 14E30 Minimal model program (Mori theory, extremal rays)
##### Keywords:
Calabi-Yau 3-folds; divisor; fibration; rational curves
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