Fano varieties with large pseudoindex and non-free rational curves

Abstract

For

, let X be a complex smooth Fano n -fold whose minimal anticanonical degree of non-free rational curves on X is at least

n − 2 $n-2$ n minus 2

. We classify extremal contractions of such varieties. As an application, we obtain a classification of Fano fourfolds with pseudoindex and Picard number greater than one. Combining this result with previous results, we complete the classification of smooth Fano n -folds with pseudoindex at least

n − 2 $n-2$ n minus 2

and Picard number greater than one. This can be seen as a generalization of various previous results. We also discuss the relations between pseudoindex and other invariants of Fano varieties.