Let X subset of P-C(N) be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m > n/2 and X is a complete intersection or that m >= N/2. We show deg(X) vertical bar deg(Y) and codim( ) Y >= codim(PN) X, where is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m >= n/2 + 1 dimensional quadrics passing through one point.