Subvarieties of small codimension in smooth projective varieties

Let X PN Cbe an n-dimensional nondegenerate smooth projective variety containing an mdimensional subvariety Y.Assume that either mn/2 and X is a complete intersection or that m N2.We show deg(X)|deg(Y)and codim Y Y codimPN X,where 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[n/2]+1 dimensional quadrics passing through one point.