Numerical characterisations of hyperquadrics

Abstract

Smooth quadric hypersuraces in $\mathbb{P}^{n+1} (\mathbb{C})$ are numerically characterised as the smooth Fano $n$-folds of length $n$, i.e., a smooth Fano $n$-fold $X$ is isomorphic to a hyperquadric if and only if the minimum of the intersection number $(C, -K_X)$ is $n$, where $C$ runs through the rational curves on $X$.