Subvarieties of small codimension in smooth projective varieties

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Let Xⁿ 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)|deg(Y) and codimspan(Y)Y≥ codimₚⁿX, 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≥[n/2]+1 dimensional quadrics passing through one point.