Subvarieties of small codimension in smooth projective varieties

Abstract

Let X ⊊ ℙ ℂ 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.