Subvarieties of small codimension in smooth projective varieties

Let X\subsetneq$\mathbb{P}${$\mathbb{C}$}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim{span(Y)}Y\geq codim_{$\mathbb{P}$^{N}}X, where span(Y) 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\geq[\frac{n}{2}]+1 dimensional quadrics passing through one point.