Numerical characterisation of quadrics

TL;DR

This paper characterizes Fano manifolds with all rational curves having high anticanonical degree, proving they are either projective spaces or quadrics, thus classifying such manifolds based on their rational curves.

Contribution

It provides a classification of Fano manifolds with a specific rational curve degree condition, identifying them as projective spaces or quadrics.

Findings

Fano manifolds with all rational curves of degree at least dimension are projective spaces or quadrics.

The result narrows down the structure of such manifolds based on rational curve properties.

The proof relies on properties of rational curves and their degrees in Fano manifolds.

Abstract

Let X be a Fano manifold such that every rational curve in X has anticanonical degree at least the dimension of X. We prove that X is a projective space or a quadric.