Polarized varieties whose points are joined by rational curves of small degrees

Abstract

Let $X$ be a projective variety with $\mathbb{Q}$-factorial singularities, over an algebraically closed field $k$ of characteristic $0$, $L$ an ample Cartier divisor on $X$, and $x$ a non-singular point of $X$. We prove that if for two general points $y,z \in X$ there exists a rational curve $C$ passing through $x, y, z$ such that $(L.C) = 2$, then $(X.L) \simeq (\mathbb{P}^{n}.\mathcal{O}(1))$ or $(Q^{n}.\mathcal{O}(1))$, a hyperquadric.