Subvarieties of small codimension in smooth projective varieties

Let X ⊊ ℙℂN be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m > \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{n} {2} $\end{document} and X is a complete intersection or that m ⩾ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{N} {2} $\end{document}. We show deg(X) | deg(Y) and codim〈Y〉Y ⩾ codimℙNX, where 〈Y〉 is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m ⩾ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\left[ {\frac{n} {2}} \right] $\end{document} + 1 dimensional quadrics passing through one point.