POLARIZED VARIETIES WHOSE POINTS ARE JOINED BY RATIONAL CURVES OF SMALL DEGREES

摘要

Let X be a projective variety with Q-factorial singularities, over an algebratically 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 implied by X there exists a rational curve C passing through x,y,z, such that (L.C) =2, then (X.L) approx=(P~n,O(1)) or (Q~n,O(1)),a hyperquadric.