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Existence of good divisors on Mukai varieties. (English) Zbl 0970.14023
This paper is related to the classification problem for Fano $$n$$-folds $$X$$ of index $$i$$. It is known that if $$i\geq n$$, $$X$$ is either a quadric or a projective space; Fano $$n$$-folds of index $$i=n-1$$ have been classified by Fujita, Campana-Flenner, Sano; Mukai gave a classification of the smooth Fano $$n$$-folds of index $$i=n-2$$ (the “Mukai varieties”) under the assumption that the linear system $$|H|$$ such that $$-K_X=(n-2)H$$ contains a smooth divisor.
In this note the author proves that such a hypothesis is always satisfied. More precisely he shows that a Mukai variety with at most log terminal singularities has, except two special cases, “good divisors”, i.e., the generic element of $$|H|$$ has at worst the same singularities as $$X$$.

##### MSC:
 14J45 Fano varieties 14C20 Divisors, linear systems, invertible sheaves 14B05 Singularities in algebraic geometry
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