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Let X ⊊ ℙ 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 ⩾ \(\frac{N}{2}\). 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 ⩾ \(\left[ {\frac{n}{2}} \right]\) + 1 dimensional quadrics passing through one point.

Subvarieties of small codimension in smooth projective varieties