## Classification of logarithmic Fano threefolds.(English)Zbl 0658.14019

Throughout this paper varieties are defined over a fixed algebraically closed field of characteristic zero. A logarithmic Fano threefold is defined to be a pair (V,D) of a smooth projective threefold V and a reduced divisor D with normal crossings on V, satisfying the following condition: $$-K_ V-D\quad is\quad ample.$$
The purpose of this paper is to classify logarithmic Fano threefolds (V,D) with non-zero boundaries D. Fundamental tools are Norimatsu’s vanishing theorem, Tsunoda’s logarithmic cone theorem, Mori’s theory of extremal rational curves on a threefold and some ampleness criteria for the logarithmic anti-canonical divisor, which will be explained in section 1.
In section 2 we study some general properties of logarithmic Fano varieties (V,D) of arbitrary dimension.
In section 4 we prove the existence of an extremal rational curve $$\ell$$ with $$(D\cdot \ell)>0$$ as a key lemma. Moreover all the types of $$\ell$$ are F, $$E_ 2, D_ 3, D_ 2$$ or $$C_ 2$$. - Logarithmic Fano threefolds (V,D) with $$D\neq 0$$ are classified into five types.
We give in section 5 the configuration of boundaries.
From sections 6 to 9 we classify logarithmic Fano threefolds according to the types of extremal rational curves.

### MSC:

 14J30 $$3$$-folds 14J10 Families, moduli, classification: algebraic theory
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### References:

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