Defining line bundle of a properly embedded hypersurface
= Defining line bundle of a properly embedded hypersurface
Local defining functions $f_\alpha$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_\alpha/f_\beta$ on overlaps. These ratios define a <real line bundle> with a global transverse section whose local representatives are the $f_\alpha$ and whose zero set is $Y$. The bundle is naturally isomorphic to the <normal bundle> of $Y$.