Fano varieties with large pseudoindex and non-free rational curves

TL;DR

This paper classifies certain Fano varieties with large pseudoindex and non-free rational curves, extending previous classifications and exploring relationships between invariants.

Contribution

It provides a classification of extremal contractions for Fano varieties with large pseudoindex and non-free rational curves, and completes the classification for cases with higher Picard number.

Findings

Classification of extremal contractions for these Fano varieties

Complete classification of Fano n-folds with pseudoindex ≥ n-2 and Picard number > 1

Relations between pseudoindex and other invariants of Fano varieties

Abstract

For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. 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 $n-2$ 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.