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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 \(X\).

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